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ARTICLES A Density Functional Study on Transition-Metal-Coated Single-Walled Carbon Nanotubes Guo Wang and Yuanhe Huang* Department of Chemistry, Beijing Normal UniVersity, Beijing 100875, China ReceiVed: NoVember 13, 2007; ReVised Manuscript ReceiVed: March 23, 2008
A density functional study on transition-metal-coated single-walled carbon nanotubes shows that there are both magnetic and nonmagnetic materials, depending on the nature of the transition metals and their adsorption sites. Adsorption of transition metal atoms to a nanotube is energetically favorable. It is found that some of the magnetic materials exhibit large magnetic moments and high spin polarizations, whereas nonmagnetic materials can display high superconducting transition temperatures. 1. Introduction Single-walled carbon nanotubes (SWNTs) were first discovered in arc discharge from carbon rods containing transition metal (TM) catalysts. 1,2 By mixing TMs with SWNTs in varying proportions, one can tune the electronic and magnetic properties. Moreover, experiments 3,4 have shown that TMs such as titanium can form a continuous coating on the SWNTs. A recent calculation has demonstrated that either a chromium or a vanadium atomic chain adsorbed on an armchair SWNT can open up a band gap for minority spin and make the whole system a complete spin-polarized conductor. 5 Spintronics 6,7 is a fast-developing field because information may be carried by two spin orientations of conduction electrons. The adsorbed TMs on carbon nanotubes may form one-dimensional (1D) nanomaterials of interest for spintronics. Theoretical calculations have been carried out for TM atomic chains adsorbed on SWNTs 5,8–13 and for titanium-coated SWNTs.14,15 Further detailed and systematic research is still needed to provide a theoretical foundation for real applications of these new nanomaterials. Herein we investigated SWNTs coated with the common first row (3d) TMs in various adsorption patterns. One of our aims is to see which TM is most likely suitable for spindependent transport. Moreover, we are interested in the possibility of forming 1D high temperature superconductors for the TM-coated SWNTs. The superconducting transition temperature (Tc) of the pristine SWNT is found to be below 1 K,16 whereas the TMs can exhibit higher Tc values (about several K) due to strong electron-phonon (e-p) coupling.17 How about the Tc of TM-coated SWNTs? In the present work, we have performed density functional calculations on the TM-coated SWNTs. The calculations showed large magnetic moments and high spin polarizations in several TM-coated SWNTs, and high Tc values for nonmagnetic structures were obtained. 2. Method and Computational Details A series of 3d TMs (Sc, Ti, V, Cr, Mn, Fe, Co, and Ni) is chosen for adsorbing on an armchair SWNT (4, 4). Four * To whom correspondence should be addressed. E-mail: yuanhe@ bnu.edu.cn.
Figure 1. Four adsorption patterns. T: above the C atom (top site), H: above the center of the hexagon (hole site), B1 and B2: above the C-C bond (bridge sites).
adsorption patterns are investigated (see Figure 1). For H and B1 cases, all the sites are occupied by TMs. The TMs are alternately placed on the C atoms in the T pattern in order to avoid overaccumulation. As for B2, the TMs are also alternately distributed along the tube axis direction. As a result, only 50% of T or B2 sites are occupied. With this distribution of TMs, the translation symmetry of the original SWNT is retained. Moreover, the TM encapsulated SWNT may be viewed as a hybrid structure in which each unit cell contains 4n C atoms and 2n TM atoms for the (n, n) SWNT. The TM distribution densities on the SWNT surface are all the same for the four adsorption patterns. The calculations are carried out using the generalized gradient approximation (GGA) PW91 density functional18,19 and the ultrasoft pseudopotential20 plane wave method implemented in VASP code.21–24 The ultrasoft pseudopotentials used are generated at the same GGA level. We set the lattice constants a ) b ) 20 Å and c ) cSWNT, where cSWNT is the 1D lattice constant of the original SWNT. The structural parameters, as well as the lattice constants, are fully relaxed in spin-polarized calculations. A (1, 1, 21) k-point sampling is used in the first Brillouin zone within the Monkhorst-Pack special k-point scheme25 in order to obtain the electronic and magnetic properties. Both relaxed lattice constants a and b are close to the original value (20 Å), hence the full optimization does not affect the 1D characteristics of the hybrid structures. 3. Results and Discussion The optimization shows that the minimum distance between the TM and C atoms are in a range of 1.97-2.50 Å (see Table 1), suggesting a covalent TM-SWNT interaction. The TMs
10.1021/jp7108253 CCC: $40.75 2008 American Chemical Society Published on Web 05/30/2008
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TABLE 1: Bond lengths (d) between TMs and the nearest C atoms, binding energies (Eb) per TM, magnetic moments (µ) per TM, total density of states (DOS) per eV per cell at the Fermi level, and spin polarizations (P) of TM-coated SWNT TM
site
d (Å)
Eb (eV)
µ ( µB )
DOS
P
Sc
T H B1 B2 T H B1 B2 T H B1 B2 T H B1 B2 T H B1 B2 T H B1 B2 T H B1 B2 T H B1 B2
2.33 2.48 2.47 2.41 2.13 2.35 2.28 2.28 2.07 2.36 2.23 2.20 2.11 2.37 2.25 2.26 2.15 2.49 2.14 2.24 2.03 2.50 2.08 2.14 1.99 2.46 2.00 2.00 2.00 2.19 2.00 1.97
2.84 2.72 2.26 2.48 2.94 2.93 2.68 2.76 2.67 2.19 2.15 2.20 0.96 0.77 0.40 0.91 1.15 -1.03 0.84 1.13 2.87 1.88 2.32 2.32 3.00 1.89 2.51 2.48 3.06 2.21 2.68 2.68
0.0 0.6 1.6 0.0 0.7 1.4 2.5 1.0 1.9 3.1 3.4 3.1 4.2 4.2 3.4 4.4 4.0 0.0 4.1 4.2 2.8 3.2 2.9 3.0 1.5 1.7 1.7 1.4 0.3 0.0 0.0 0.2
14.6 19.1 12.7 19.9 12.0 19.1 17.3 22.6 13.7 13.9 12.5 20.5 15.0 12.2 12.6 10.7 14.3 27.0 14.7 15.7 14.9 20.2 13.6 15.7 14.4 19.7 19.7 20.0 14.2 11.5 14.0 16.8
0.00 0.49 0.35 0.00 0.06 0.51 0.47 0.40 0.36 0.44 0.41 0.03 0.50 0.63 0.08 0.33 0.54 0.03 0.82 0.53 0.63 0.72 0.63 0.68 0.86 0.67 0.68 0.61 0.44 0.00 0.00 0.50
Ti
V
Cr
Mn
Fe
Co
Ni
adsorbed to the T site deviate from the sites and dimerizations occur (Figure 2), especially for Sc, V, Fe, Co, and Ni. The minimum TM-TM distance is 2.37 Å for Co. Define the binding energy (Eb) per TM as
Eb ) (nETM + ESWNT - ETM@SWNT) ⁄ n The calculated binding energies (see Table 1) are all positive (0.40-3.06 eV) with only one exception in the case of Mn at H site (it is not a stable structure and will not be discussed later), indicating that the TM-coated SWNTs are energetically favorable. An Eb plot for different TMs at different adsorption sites is shown in Figure 3. It is found that, from Sc, the Eb of all sites decrease gradually until Cr or Mn, and then increase up to Ni, which indicates that the TMs with half-filled d-shell have less affinity to the SWNT. Furthermore, adsorption to T site is always most favorable to the stability of the hybrid
Figure 2. Dimerizations of TMs. (left) Frontal view and (right) lateral view.
Figure 3. Binding energies of different TMs at different adsorption sites.
systems. This is related to the dimerizations at T site described above, which leads to additional stabilities obtained from the increase of the TM-TM interaction along the circumference. The calculated band shape of the hybrid structures depends on the TMs, spins, and adsorption patterns. However, there always exist many partially filled bands, so these systems are all metals. In fact, even the Ti-coated (8, 0) SWNT 14,15 has these characteristics, although the pristine (8, 0) is a semiconductor. As an example, the band structures in the case of Co adsorbed to the T site are given in Figure 4, and this configuration has a large Eb and a high spin polarization. For comparison, the band structures of the corresponding TM tube and the pristine (4, 4) SWNT are also given. It is found that the characteristics of the SWNT frontier bands are no longer retained in the band structures of the hybrid system, which are similar to those of the corresponding TM tube. A detailed analysis shows that the partially filled bands or conduction bands are overwhelmingly contributed from the d orbitals of the TMs (for instance, 84% in the case of Co at the T site). The rest are contributed from the p orbitals of the C atoms, followed by the p and s orbitals of the TMs. Because of the existence of several partially filled bands, large density of state (DOS) at the Fermi level is obtained (see Table 1). The DOS of the hybrid systems ranges from 1.0 to 17.5 states per eV per cell per spin, 6.5-108.8 times as large as that of the pristine SWNT. Local DOS and projected DOS analyses show that the d orbitals of the TMs make an overwhelming contribution to the total DOS at the Fermi level. For example, in the case of Co adsorbed to the T site, 90% of the total DOS at the Fermi level is contributed from the d orbitals of the TMs and 4% is from the p orbitals of the C atoms. Figure 5 shows the total DOS and the DOS contributed from the d orbitals of the Co atoms, which indicates that the d orbitals of the Co atoms mainly contribute to the total DOS near the Fermi level. The results of the local and projected DOS analyses conform to the above conduction band analysis. Total magnetic moments and spin polarizations at the Fermi level for each hybrid structure are also listed in Table 1. Large magnetic moments are found for Cr, Mn, V, and Fe-coated SWNTs, followed by Co, Ti, and Sc-coated ones. The largest one is 4.4 µB per TM in the case of Cr adsorbed to the B2 site. Ni-coated SWNTs are almost nonmagnetic, similar to the theoretical calculations in the case of one Ni atom adsorbed on SWNTs.26–28 The magnetic moment increases from Sc to Cr (or Mn), and then decreases to Ni in all adsorption patterns.
9130 J. Phys. Chem. C, Vol. 112, No. 25, 2008
Wang and Huang
Figure 4. Band structures of the Co-coated SWNT (T configuration) for (a) majority and (b) minority spin, the corresponding Co tube for (c) majority and (d) minority spin, (e) (4, 4) SWNT, and (f) the Sc-coated SWNT (T configuration) near the Fermi level (set to zero).
Figure 6. (a) Total charge density and (b) charge density isocharge surface for the difference between the majority and minority spin in the case of Co adsorbed to the T site.
Figure 5. DOS in the case of Co adsorbed to the T site. (The solid curve represents the total DOS and the filled area under the curve represents the contribution from the d orbitals of the Co atoms. The Fermi energy is set to zero.)
This site-irrelevant variation trend of the magnetic moment is consistent with that of the free TM atoms. In the case of the same TM adsorbed to different sites, the magnetic moments have some differences. For Sc, Ti, and V, the difference is about 1-2 µB, but for Cr and heavier TMs, the difference becomes smaller. Because of their large magnetic moments, Cr, Mn, V, and Fe-coated SWNT would be 1D magnetic materials if uniform coated structures are available. Besides, the variation trend of the magnetic moments is just opposite to that of the Eb. A large magnetic moment is accompanied by small Eb. The reason would be complex. One explanation is that the TMs maintaining large magnetic moments will have smaller electronic structural changes from their free atom states and will then have weaker interactions between the C atoms, so the Eb is smaller. For Mn, the free atom has a magnetic moment of 5.0 µB according to the Hund’s rule. The drop of the magnetic moment is 1.0 µB for the most stable T configuration after adsorption to the SWNT, whereas in the case of Ni at T site, the drop is 1.7 µB, from 2.0 to 0.3 µB. The Eb of the Ni-coated SWNT is larger than that of the Mn-coated SWNT for the T configuration. The electron rearrangement may be one factor that makes the systems stable. As for Cr and Mn-coated SWNTs with high magnetic moments but small Eb, experimental technologies can make the hybrid systems stable, such as introducing a Ti buffer layer between the TMs and the SWNT, which was developed by Zhang and Dai.4 Further investigation on the magnetic properties of these systems is in progress. The spin polarization is defined as the absolute value of [N v(EF) - N V(EF)]/[N v(EF) + N V(EF)], where N v(EF) [NV(EF)]
represents the DOS of majority (minority) spin at the Fermi level. If the orbital at the Fermi level of a ferromagnet has d characters, the spin polarization will be high.29 From the above conduction band and projected DOS analyses, the d orbitals of the TMs overwhelmingly contribute the total DOS at the Fermi level. As a result, this type of materials may be a candidate for high spin polarization materials. Our calculations show that some TM-coated SWNTs exhibit high spin polarizations, especially in the case of Co at the T site (P ) 0.86) and Mn at the B1 site (P ) 0.82). The imbalance of majority and minority spin at the Fermi level will produce a spin-dependent transport process. Traditional Fe, Co, and Ni materials have spin polarizations of 40-50%.29 Materials with higher spin polarizations would offer a higher magnetoresistance ratio and would be more favorable for spin-dependent transport. However, the values of the spin polarizations differ from different TMs and different adsorption sites. This may be due to the different magnetic properties of the TMs and the complex interactions between these TMs and the SWNT, but Fe and Co-coated SWNTs always possess large spin polarizations (P > 60%) for all sites considered. It is interesting that Fe and Co maintain their spin polarizations when coated to SWNT but Ni does not. Maybe the stronger interaction makes Ni-coated SWNTs have lower magnetic moments and not high spin polarizations. As calculated by Yang and coworkers,30 Fe and Co inside the SWNT also possess large spin polarizations (the largest P is 0.87). These results show that Fe and Co will have constant large spin polarizations when interacting with SWNTs. Meanwhile, they have considerable magnetic moments, 2.8 and 1.5 µB in the case of the T site, respectively. Hence, Fe and Co are likely to be natural building blocks for 1D spin-dependent transport when interacting with SWNTs. Total charge density of Co at T site is shown in Figure 6a. It is observed that the charge is distributed at both the C and the TM atoms. The isocharge surface for the difference between the majority and minority spin charge density (shown in Figure
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6b) reveals that the spin polarization is localized at the TM atoms, and this concentrated spin polarization is responsible for the magnetic moment. Our calculations show that some Sc and Ni-coated SWNTs are nonmagnetic. Although nonmagnetic structures are not suitable for spin-dependent transport, it is worth exploring the superconduction, considering the large DOS at the Fermi level. To obtain the symmetry of the electron wave function, which is needed to determine the possible coupling with a certain phonon mode, we carry out self-consistent-field crystal orbital (SCF-CO) calculations using a linear combination of atomic orbital-based program CRYSTAL06.31 Here we take the case of Sc adsorbed to the T site as an example, and the calculated band structure is shown in Figure 4f. This configuration is the most stable one of the Sc-coated SWNTs, and it has zero magnetic moment. The (4, 4) SWNT is also calculated for comparison. The computation is performed with B3LYP 32,33 functional and 6-21G* and 864-11G* basis set for C and Sc, respectively.34 Default values of convergence criteria in the CRYSTAL06 program are used. The SCF convergence criterion is set to 10-6 Hartree; 55 radial and 434 angular points are used for atomic grid in order to evaluate the density functional numerical integrations, and the shrinking factor is set to 20. Under e-p coupling mechanism, the intracell e-p coupling constant can be calculated by eq 135
λintra ) N(EF)
∑ Vn
(1)
n
where N(EF) is the DOS per eV per cell at the Fermi level, ∑ Vn is the sum of e-p interaction potential Vn for all phonon n modes with the same symmetry. Here, Vn is calculated numerically by variation of the Fermi energy with respect to the small variation (0.3%) of the normal mode Qn at the Γ point 36,37
Vn )
(∂ε ⁄ ∂Qn)2 pωn
(2)
in which ωn is the phonon frequency of Qn. First, we have calculated the e-p coupling constant of (4, 4) SWNT which has C4V symmetry under the periodic boundary condition in the SCF-CO calculations. The electron wave function and the phonon mode should have the same symmetry in order to obtain a nonzero coupling for this symmetry group. The two frontier bands have A1 and A2 symmetries, respectively. There are six phonon modes for A1, A2, B1, and B2 symmetries, whereas there are twenty-four E modes. The calculated DOS is 0.53 state per eV per cell. The e-p coupling constant for the A1 (A2) mode is 0.046 (0.049), close to the value 0.062 used for (5, 5) SWNT in the reference,38 which is obtained from the parameters extracted from a 2D graphite sheet.39,40 We also calculated the intercell e-p coupling constant under the deformation potential (DP) theory.41 The intercell e-p coupling constant λinter can be expressed as eq 3,42
λinter)
N(EF)ε12 MνF2
(3)
where VF ) (2π/h)(dE/dk)|k)kF is the Fermi velocity, 1 is the proportionality constant of the DP, and M is the mass per cell. The variation of the energy at the Fermi level is linearly related to the small variation of the lattice constant c; that is, δ ) 1∆ and ∆ ) δc/c. The calculated λinter is only about 10-8. The total coupling constant can be calculated by λtotal ) λintra + λinter. Because λintra is far larger than λinter, the superconducting
properties should be mainly determined by the intracell e-p coupling. The transition temperature can be calculated with the classical BCS Tc formula (eq 4),43
Tc)1.14ω exp(-1/λ)
(4)
where ω ) 1/2 here. The strong C-C bonds lead to large 1/2, 1254 K (1167 K) for the A1 (A2) mode. The calculated Tc is 5.2 × 10-7 K (1.8 × 10-6 K) for the A1 (A2) mode, far less than 1 K. The weak e-p coupling leads to a very small Tc, although the phonon frequency is relatively high. As in the case of Sc adsorbed to the T site, the crystal orbitals of the frontier bands have A, B, and E symmetries of the C4 symmetry group, and there are 18 A, 18 B and 36 E phonon modes. The calculated intracell e-p coupling constant is 76.1, 106.9, and 164.4 for A, B, and E symmetry, respectively. The intercell coupling (about 10-8) can also be neglected. Because the e-p coupling constant is larger than 10, we use eq 5,
Tc)0.15(λ)1/2 Dynes.17
(5) 1/2
developed by Allen and The calculated is 747, 590, and 692 K for A, B, and E modes, respectively. Then, the Tc is 977, 915 and 1331 K for A, B, and E modes, respectively. These values should be considered qualitative only, but compared with that of the SWNT, the Tc significantly increases. The 1D TM-coated SWNT displays the possibility of high Tc. The calculations show that the average ∂ε/∂Qn is about 4 times as large as that of (4, 4) SWNT, indicating stronger e-p interaction of the d electrons with the vibration modes. The e-p interaction potential should be 42 times as large as that of the pristine SWNT according to eq 2. Besides the enhanced e-p interaction potential, the DOS of the Sc-coated SWNT is 12 times as large as that of (4, 4) due to the concentrated d bands at the Fermi level. Moreover, the average frequencies of 1/2 are only somewhat smaller than those of A1 and A2 modes of the pristine SWNT, which indicates that the frequency change has very limited influence on the Tc of the TM-coated SWNT. Therefore, it is the large e-p interaction potential and the large DOS that lead to strong e-p coupling and high Tc. It is noticed that the B3LYP SCF-CO calculated DOS (6.2 states per eV per cell) is smaller than that of the plane wave calculation with a different density functional PW91, but it does not change the qualitative nature of the high T c. Because the distribution of the TMs on the SWNT surface may be complex and the symmetry may be different from this C4 symmetry in the experiment, we have also calculated the average e-p coupling constant λj of each phonon mode for comparison. The calculated λj is 5.1, much smaller than the above λ (larger than 76), but is still big. Because λj is between 1.5 and 10, we use the corrected McMillan Tc formula (eq 6)17,44 and assume the effective Coulomb repulsion parameter µ* ) 0.1, as in the reference.17
Tc)( f1 f2ωlog /1.20)exp[-1.04(1 +λ)/(λ-µ* - 0.62λµ*)] (6) f1 is a strong-coupling correction factor, and f2 is a shape correction factor. With the ωlog ) exp [∑ ((λn ln ωn)/λ)] ) n 365K, the calculated Tc is 155 K. It can be seen that high Tc is obtained even with the smaller average e-p coupling constant. This further indicates that the TM-coated SWNT may be a candidate for high temperature superconductors, from a view of theoretical calculations. In fact, the nonmagnetic hybrid
9132 J. Phys. Chem. C, Vol. 112, No. 25, 2008 structures inherit the properties in favor of superconduction from the two constituents, the TM and the SWNT. The TMs have large e-p coupling constants (about 1 in bulk form),17 and this property is even enhanced when coated to the SWNT. Likewise, the SWNT has high frequencies, and this property has only a little decrease in the hybrid system. Therefore, the new structure shows high Tc values and provides us with a new area for seeking high temperature superconduction materials. Although we have only considered the e-p coupling mechanism and the calculations can still be viewed qualitatively, compared with the very low Tc for the pristine SWNT,16,38 it is true that the Tc significantly increases. Large DOS, strong e-p interaction, and high phonon frequency of this hybrid structure result in high Tc values. The nonmagnetic TM-coated SWNT may be a suitable nanomaterial for high temperature superconductors. 4. Conclusions In summary, we have studied TM-coated (4, 4) SWNTs using density functional theory. Large magnetic moments and high spin polarizations are found in these materials. Fe and Co-coated SWNTs may be candidates for 1D spin-dependent transport. Moreover, calculations show strong electron-phonon coupling in the nonmagnetic TM-coated SWNT, which results in high superconducting transition temperature. Therefore, the TMcoated SWNT may also be a candidate for 1D superconduction materials. TM-coated SWNTs with larger diameter, such as TMcoated (5, 5) and (6, 6), may have similar properties if the same adsorption patterns are applied. Further investigation is in progress. Acknowledgment. This work is supported by RFDP (20060027001), the Major State Basic Research Development Program (Grant No. 2002CB613406), and NSFC (20373008). References and Notes (1) Iijima, S.; Ichihashi, T. Nature 1993, 363, 603. (2) Bethune, D. S.; Kiang, C.-H.; de Vries, M. S.; Gorman, G.; Savoy, R.; Vazquez, J.; Beyers, R. Nature 1993, 363, 605. (3) Zhang, Y.; Franklin, N. W.; Chen, R. J.; Dai, H. Chem. Phys. Lett. 2000, 331, 35. (4) Zhang, Y.; Dai, H. Appl. Phys. Lett. 2000, 77, 3015. (5) Yang, C.-K.; Zhao, J.; Lu, J. P. Nano Lett. 2004, 4, 561. (6) Prinz, G. A. Science 1998, 282, 1660. (7) Wolf, S. A.; Awschalom, D. D.; Buhrman, R. A.; Daughton, J. M.; von Molna´r, S.; Roukes, M. L.; Chtchelkanova, A. Y.; Treger, D. M. Science 2001, 294, 1488. (8) Yang, C.-K.; Zhao, J.; Lu, J. P. Phys. ReV. B 2002, 66, 041403.
Wang and Huang (9) Durgun, E.; Ciraci, S. Phys. ReV. B 2006, 74, 125404. (10) Fagan, S. B.; Mota, R.; da Silva, A. J. R.; Fazzio, A. Microelectron. J. 2003, 34, 481. (11) Fagan, S. B.; Mota, R.; da Silva, A. J. R.; Fazzio, A. J. Phys.: Condens. Matter 2004, 16, 3647. (12) Fagan, S. B.; Fazzio, A.; Mota, R. Nanotechnology 2006, 17, 1154. (13) Jo, C.; Lee, J. I.; Jang, Y.-R. Surf. Sci. 2006, 600, 1592. (14) Dag, S.; Durgun, E.; Ciraci, S. Phys. ReV. B 2004, 69, 121407. (15) Dag, S.; Ciraci, S. Phys. ReV. B 2005, 71, 165414. (16) Kasumov, A. Y.; Deblock, R.; Kociak, M.; Reulet, B.; Bouchiat, H.; Khodos, I. I.; Gorbatov, Y. B.; Volkov, V. T.; Journet, C.; Burghard, M. Science 1999, 284, 1508. (17) Allen, P. B.; Dynes, R. C. Phys. ReV. B 1975, 12, 905. (18) Perdew J. P. In Electronic Structure of Solids, Ziesche, P.; Eschrig, H. Eds.; Akademie Verlag: Berlin, 1991. (19) Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhais, C. Phys. ReV. B 1992, 46, 6671. (20) Vanderbilt, D. Phys. ReV. B 1990, 41, 7892. (21) Kresse, G.; Hafner, J. Phys. ReV. B 1993, 47, 558. (22) Kresse, G.; Hafner, J. Phys. ReV. B 1994, 49, 14251. (23) Kresse, G.; Furthmu¨ller, J. Phys. ReV. B 1996, 54, 11169. (24) Kresse, G.; Furthmu¨ller, J. Comput. Mater. Sci. 1996, 6, 15. (25) Monkhorst, H. J.; Pack, J. D. Phys. ReV. B 1976, 13, 5188. (26) Durgun, E.; Dag, S.; Bagci, V. M. K.; Gu¨lseren, O.; Yildirim, T.; Ciraci, S. Phys. ReV. B 2003, 67, 201401. (27) Durgun, E.; Dag, S.; Ciraci, S.; Gu¨lseren, O. J. Phys. Chem. B 2004, 108, 575. (28) Yagi, Y.; Briere, T. M.; Sluiter, M. H. F.; Kumar, V.; Farajian, A. A.; Kawazoe, Y. Phys. ReV. B 2004, 69, 075414. (29) Soulen Jr, R. J.; Byers, J. M.; Osofsky, M. S.; Nadgorny, B.; Ambrose, T.; Cheng, S. F.; Broussard, P. R.; Tanaka, C. T.; Nowak, J.; Moodera, J. S.; Barry, A.; Coey, J. M. D. Science 1998, 282, 85. (30) Yang, C.-K.; Zhao, J.; Lu, J. P. Phys. ReV. Lett. 2003, 90, 257203. (31) Dovesi R.; Saunders V. R.; Roetti C.; Orlando R.; Zicovich-Wilson C. M.; Pascale F.; Civalleri B.; Doll K.; Harrison N. M.; Bush I. J.; Ph. D′ Arco; Llunell M. CRYSTAL06 User’s Manual; University of Torino: Italy, 2007. (32) Becke, A. D. J. Chem. Phys. 1993, 98, 5648. (33) Lee, C.; Yang, W.; Parr, R. G. Phys. ReV. B 1988, 37, 785. (34) CRYSTAL Home Page, http://www.crystal.unito.it (accessed April 2008). (35) Tanaka, K.; Huang, Y.; Yamabe, T. Phys. ReV. B 1995, 51, 12715. (36) Pascale, F.; Zicovich-Wilson, C. M.; Lopez, F.; Civalleri, B.; Orlando, R.; Dovesi, R. J. Comput. Chem. 2004, 25, 888. (37) Zicovich-Wilson, C. M.; Pascale, F.; Roetti, C.; Saunders, V. R.; Orlando, R.; Dovesi, R J. Comput. Chem. 2004, 25, 873. (38) Huang, Y.; Okada, M.; Tanaka, K.; Yamabe, T. Phys. ReV. B 1996, 53, 5129. (39) Pietronero, L.; Stra¨ssler, S.; Zeller, H. R. Phys. ReV. B 1980, 22, 904. (40) Jishi, R. A.; Dresselhaus, G. Phys. ReV. B 1982, 26, 4514. (41) Bardeen, J.; Shockley, W. Phys. ReV. 1950, 80, 72. (42) Huang, Y.; Chen, G.; Liu, R. J. Phys. Chem. Solids 1998, 59, 1365. (43) Bardeen, J.; Cooper, L. N.; Schrieffer, J. R. Phys. ReV. B 1957, 108, 1175. (44) McMillan, W. L. Phys. ReV. 1968, 167, 331.
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