Polyhedral transformation and phase transition in TcO₂

Abstract

By using ab initio random structure search, we have predicted a stable low pressure phase (phase-LP) of TcO₂, which shares the same space group with the ambient condition phase (phase-AC) of TcO₂ but exhibits very different crystallographic features. A novel polyhedral transformation from the TcO₆-octahedrons in phase-AC to TcO₅-hexahedrons in phase-LP has been observed in the phase transition. Large voids along the (100) and (001) directions in phase-LP protect its low density stability, which can serve as the transport channel for protons and β particles. We show that the TcO₅-hexahedrons in phase-LP are more distorted and show stronger anisotropy than the TcO₆-octahedrons in phase-AC. According to the analysis on the electronic structures and chemical bonding of TcO₂, the various origins of the metallic nature of phase-AC and phase-LP have been explored and discussed as well.

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1. Introduction

Technetium (Tc) with atomic number 43, located in the middle of the transition metals in the periodic table, is the first artificially made element with a long half-live,^1,2 which behaves like a real 4d transition metal. Tc possesses no stable isotope in the natural environment. However, man-made ⁹⁹Tc produced from nuclear reactors is a pure β-emitter, and has been widely used as a radioactive tracer in diagnostic imaging procedures in nuclear medicine.³ The 4d⁵5s² electron configuration of Tc makes it possible to form complex compounds^4–6 and alloys.^7–9 Recently, Tc based ternary perovskites CaTcO₃ (ref. 10) and SrTcO₃ (ref. 11 and 12) have been synthesized, which show distinguished magnetism with anomalously high Neél temperatures. Soon afterward, Franchini et al.¹³ discovered that the origin of the strong magnetism in Tc based perovskites is derived from the cooperative rotation of the TcO₆ octahedrons. Later in 2012, Borisov et al.¹⁴ found a positive correlation between the critical temperature and volume as well as the magnetic exchange interactions in the Tc based perovskites. Mravlje et al.¹⁵ found that the origin of the high Néel temperature of SrTcO₃ is due to the itinerant-to-localized transition in the Tc compound. These studies extend the understanding of the magnetic properties in perovskites. There exist two identified technetium oxide: black TcO₂ and yellow Tc₂O₇. Hereinto, TcO₂ is formed by the same trivalence Tc³⁺ and distorted TcO₆ octahedrons in ternary perovskites CaTcO₃ and SrTcO₃. Taking into consideration the novel features of CaTcO₃ and SrTcO₃, the study of TcO₂ is of great interest and importance as well. The structure of black TcO₂ has been firstly solved by Rodriguez et al.¹⁶ in 2007. Recently, Taylor¹⁷ explored the oxidation of Tc metal and the Tc–TcO₂ interface. However, a systematical study of the physical and chemical properties of technetium oxide is still lacking.

External pressure is a powerful tool to adjust the physical and chemical properties of materials, especially to induce the structure or property transitions.^18–20 In this work, the pressure-induced volume effect of TcO₂ has been systematically studied. A stable low pressure phase TcO₂ has been predicted by the ab initio random structure searching, such low pressure phase transition could be probably achieved by the mechanically induced buckling stress.²¹ Based on density functional theory calculations, we have systematically analyzed the crystal structure, chemical bonding, electronic structure and lattice dynamical properties for TcO₂ in the predicted low pressure phase as well as ambient condition phase.

2. Computational details

Our calculations are based on density functional theory (DFT) using the Vienna ab initio simulation package²² (VASP) in conjunction with projector augmented wave (PAW) pseudopotentials. The generalized gradient approximations²³ (GGA) of Perdew–Burke–Ernzerhof²⁴ (PBE) was used. The valence electron configuration for Tc and O were 4s²4p⁶5s²4d⁵ and 2s²2p⁴. The geometry convergence was achieved with the cut-off energy of 600 eV. K-points of 8 × 8 × 8 were automatically generated with the Γ symmetry. The relaxation convergence for ions and electrons were 1 × 10⁻⁵ and 1 × 10⁻⁶ eV, respectively. The crystal structures and polyhedrons were visualized using the VESTA²⁵ tool. The PHONOPY²⁶ code was applied to obtain the phonon frequencies within the supercell approach. We used a 2 × 2 × 2 supercell and 3 × 3 × 3 K-points for the phonon calculations.

Recently, Zhang et al. investigated perovskite CaTcO₃ using the GGA + U approach.²⁷ They pointed out that the results obtained by GGA + U provide a good description for the structural, elastic, magnetic and electronic structure of CaTcO₃. Hence we checked the effect of the Hubbard U approach to the crystallographic properties of TcO₂. The on-site Coulomb interactions in the localized Tc 5d electrons are described by using the formalism formulated by Dudarev et al.^24–26 In this scheme, the parameters U and J do not enter separately, only the effective interaction parameter U_eff = U − J is meaningful. Since in real materials, the U value tends to be about ten times larger than the J value. Setting the effective on-site Exchange interaction parameters J = 0 is not physically reasonable. We used an effective interaction parameter U_eff for the Tc 4d electrons in the range of 0 to 10 eV by keeping in mind that J = 0.5 eV has been used during the calculations. As seen from the calculated lattice parameters of TcO₂ in ambient condition phase by the GGA + U approach in Table 1, the lattice parameters a, b and c increase with the increase of U values. Meanwhile, the angle β firstly slightly decreases and then tiny increases with the increase of U values, which is not very sensitive to the U value. Comparing with the neutron powder diffraction results at the end of Table 1, the GGA + U approach does not provide better structure information. Very recently, Taylor tested GGA, GGA + U and the hybrid functional methods for the modeling of TcO₂.¹⁷ They found that GGA-PBE describes the lattice constant of TcO₂ as good as the hybrid functions, which is better than GGA + U method. Our results in Table 1 agree well with Taylor's conclusion. Hence we present the GGA-PBE calculations without Hubbard U approach in this paper, unless stated otherwise.

Table 1 The calculated lattice parameters of TcO₂ in ambient condition phase (phase-AC) by the GGA + U approach, where various U_eff values are used

Ueff (eV)	a (Å)	b (Å)	c (Å)	β (°)
0	5.663	4.793	5.560	121.27
1	5.671	4.795	5.563	121.28
2	5.683	4.796	5.567	121.29
3	5.695	4.798	5.573	121.23
4	5.698	4.810	5.588	121.12
5	5.697	4.828	5.612	121.05
6	5.704	4.842	5.628	121.05
7	5.714	4.853	5.643	121.06
8	5.728	4.864	5.658	121.13
9	5.745	4.873	5.674	121.22
10	5.765	4.879	5.687	121.26
Expt.^16 (room temperature)	5.690	4.755	5.520	121.45
Expt.^16 (15 K)	5.686	4.752	5.516	121.47

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3. Results and discussions

From the crystallographic point of view, TcO₂ in ambient condition phase (phase-AC) can be described as a distorted rutile with lattice parameters a = 5.690 Å, b = 4.755 Å, c = 5.520 Å and β = 121.45° at room temperature,¹⁶ which belongs to the space group no. 14 of P2₁/c. Fig. 1(a) illustrated the crystal structure for TcO₂ in phase-AC. As can be seen, the distorted TcO₆ octahedrons are edge-sharing concatenated along the [100] direction to form the octahedron-chains. Meanwhile, the one dimension chains are corner-sharing connected to form the three dimensional crystal.¹⁶ Our GGA-PBE-relaxed lattice parameters for TcO₂ in phase-AC are a = 5.663 Å, b = 4.793 Å, c = 5.560 Å and β = 121.27°, agree well with the experimental results within less than 1% mismatch. Since the difference between a and c is smaller than 0.1 Å and the angle β is very close to 120°, the phase-AC lattice can be considered as a distorted hexagonal lattice. When comparing the optimized atomic positions of TcO₂ in phase-AC with that of the experiments¹⁶ (given in Table 2), we observe that the results are in very good agreement with the experiments as well.

Fig. 1 (a) The crystal structure for TcO₂ in ambient condition phase (phase-AC), the large gray balls indicate the Tc atoms, the small red balls present the O atoms and the blue spaces show the distorted TcO₆ octahedrons. (b) The first Brillouin zone (1BZ) and related high symmetry points for monoclinic crystal.

Table 2 The calculated atomic positions and Bader charge Q_B (e) of TcO₂ in phase-AC and phase-LP, the experimental results are listed as a reference

Phase	Atom	x	y	z	QB
Phase-AC 0 GPa expt.^16	Tc	0.2621	0.0073	0.9844	—
	O1	0.1037	0.1883	0.195	—
	O2	0.3919	0.7085	0.2704	—
Phase-AC 0 GPa calc.	Tc	0.2655	0.0047	0.9857	13.10
	O1	0.1062	0.1925	0.1937	6.97
	O2	0.3922	0.7133	0.2690	6.92
Phase-LP −13.3 GPa calc.	Tc	0.3063	0.9402	0.0580	13.25
	O1	0.0347	0.0464	0.1860	6.80
	O2	0.4601	0.6982	0.3201	6.94

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The enthalpies vs. pressure curves of TcO₂ in phase-AC and ab initio random structure searching predicted low pressure phase (phase-LP) are shown in Fig. 2(a). One can see that phase-AC is the energetic stable structure under positive external pressure and under negative pressure above 13.3 GPa. Meanwhile, phase-LP is more stable than phase-AC below −13.3 GPa. The relaxed equilibrium lattice parameters of TcO₂ in phase-LP at −13.3 GPa are a = 5.329 Å, b = 6.176 Å, c = 6.424 Å and β = 115.33°, respectively. The optimized atomic positions of TcO₂ in phase-LP at −13.3 GPa are given in Table 2 as well. Remarkable movement of the Tc atom and the O₁ y coordinate is corresponding to the TcO₆-octahedron to TcO₅-hexahedron transition from phase-AC to phase-LP. The crystal structure of our predicted low pressure phase (phase-LP) TcO₂ is illustrated in Fig. 2(b), which shows the same space group of P2₁/c as phase-AC. There are three crystallographic key features to distinguish phase-LP from phase AC. Firstly, the angle β between axis a and c is smaller than 120° in phase-LP, which is larger than 120° in phase-AC. Secondly, the distorted TcO₆ octahedrons in phase-AC are no longer exist in phase LP and replaced by distorted TcO₅ hexahedrons. Such TcO₅ hexahedrons are edge-sharing concatenated along the [100] direction to form the hexahedron-chains. Last but not the least, the corner-sharing point of the polyhedron-chains in phase-AC and phase-LP is different. More details about the polyhedral geometric constructions of phase-AC and phase-LP will be discussed later.

Fig. 2 (a) The enthalpy as a function of pressure for TcO₂ in phase-AC and phase-LP. (b) The crystal structure for TcO₂ in phase-LP, the large gray balls indicate the Tc atoms, the small red balls present the O atoms and the blue spaces show the distorted TcO₅ hexahedrons. (c) The lattice parameters as a function of cell volume for TcO₂.

To further analysis the phase transition, we plotted the lattice parameters as a function of cell volume for TcO₂ in Fig. 2(c). The light blue part indicates the stable volume range for phase-AC, and the light violet part presents the stable volume range for phase-LP. For phase-AC, the equilibrium volume at ambient condition is 129.0 ų. There has been a gradual decrease of the lattice constants a, b and c by reducing the cell volume with positive pressures, and vice versa by enlarging the cell volume with negative pressures. The opposite tendency is showing for the angle β. The lattice of phase-AC will crash as the cell volume is further reduced to larger than 147.2 ų. By further increasing the cell volume, TcO₂ will transform to phase-LP with significant variations of the lattice parameters. The figure shows that the lattice parameter a, and angle β are decreasing and b is increasing obviously during the phase transition. As a result of the lattice variation, the phase-LP lattice cannot be considered as a distorted hexagonal lattice any more. It is worth noting that although the cell is much larger than the equilibrium volume of phase-AC, phase-LP is under positive compressive pressure from 5.1 GPa to 0 GPa in the volume range from 147.2 to 156.0 ų. The equilibrium volume at 0 GPa pressure is 156.0 ų for phase-LP, which is 20.9% larger than phase AC. For phase-LP, there is a steady increase of the lattice constants a, b and c with the increase of cell volume in the whole pressure range, which is similar to phase-AC. Nevertheless, the angle β peaks at 116.66° at 0 GPa pressure for phase-LP, which decreases regardless with the sign of the pressure.

Table 3 lists the calculated bond length for Tc–Tc and Tc–O in TcO₂ in phase-AC and phase-LP. Our calculated bond length for phase-LP agrees well with Rodriguez et al.'s experimental and theoretical results.¹⁶ Herein, the Tc–Tc bonds dose not contribute to the metallic electronic nature of TcO₂ in phase-LP,¹⁶ which can be regarded as the distance between two edge-sheared polyhedral center. Although the cell volume of phase-LP is much larger than phase-AC, we found that TcO₂ in phase-LP and phase-AC exhibit very similar polyhedral center distance by comparing the Tc–Tc bond length in Table 3. The Tc–O bond lengths span a much narrower distribution range from 1.943 to 2.034 Å in phase-AC, which distributes from 1.824 to 2.193 Å in phase-LP. This leads to the TcO₆-octahedron in phase-AC more regular than the TcO₅-hexahedron in phase-LP. The enormous distorting of the TcO₅-hexahedron in phase-LP protects the stability of the low density phase. Fig. 3 illustrated the polyhedral framework models for TcO₂ in phase-AC and phase-LP along different directions. As can be observed, both the TcO₆-octahedrons and TcO₅-hexahedrons form 3D hollow frameworks in TcO₂. For phase-AC, large voids can be found along the (100) direction only. The TcO₆-octahedral framework fills up the space along the (010) and (001) directions. For phase-LP, the voids along the (100) direction are much bigger. In addition, similar large voids can be found along the (001) direction. Such voids can serve as the transport channel for protons and β particles.

Table 3 The calculated Tc–Tc and Tc–O bond lengths (Å) of TcO₂ in different phases

Bond	Phase-AC 0 GPa	Phase-LP −13.3 GPa
Tc–Tc1	2.577	2.584
Tc–Tc2	3.092	3.111
Tc–O1	2.034	2.055
Tc–O2	2.011	2.193
Tc–O3	2.009	2.136
Tc–O4	1.988	1.966
Tc–O5	1.968	1.824
Tc–O6	1.943	—
Tc–Oavg	1.992	2.035

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Fig. 3 The polyhedral framework models for TcO₂ in (a) phase-AC and (b) phase-LP along (100), (010) and (001) directions.

The charge transfer between Tc and O atoms in TcO₂ can be viewed by the so-called Bader charge analysis,^28,29 listed in Table 2. According to the pseudopotential valence electron configurations, Tc and O atoms have original 15 and 6 valence electrons. As shown in Table 2, the ionic charges in TcO₂ can be presented as Tc^1.90+(O^0.95−)₂ for phase-AC and Tc^1.75+(O^0.875−)₂ for phase-LP, respectively. Hence, the average electron transferred from Tc to each O atom is 0.32 e in TcO₆-octahedrons for phase-AC and 0.35 e in TcO₅-hexahedrons for phase-LP, respectively. There shows no distinguished charge transfer difference for phase-AC and phase-LP. For a deeper understanding on the chemical bonding character of TcO₂, the topological analysis of the electron localization function (ELF)³⁰ has been carried out, which gives a rather quantitative description on the binding states³¹ and provides a straightforward chemical picture³² of the polyhedrons in TcO₂. The ELF line profile from Tc to O for the Tc–O bonds in the TcO₆ octahedrons of phase-AC and TcO₅ hexahedrons of phase-LP are illustrated in Fig. 4(a) and (b), respectively. The relative bond strength can be characterized by the so-called bond point corresponding to the saddle point in the ELF with two negative eigenvalues and one positive eigenvalue of the Hessian matrix.³³ Herein, the minimum ELF value in the middle of the curve reflects to the bond point.³¹ As seen in Fig. 4(a), the Tc–O bonding in phase-AC is rather homogeneous. The minimum ELF value of Tc–O₁ (0.165) is very close to the maximum value of Tc–O₆ (0.195), which indicates very small distortion of the TcO₆ octahedrons. On the other hand, as seen from Fig. 4(b), the ELF values for Tc–O₁, Tc–O₂, Tc–O₃, Tc–O₄ and Tc–O₅ are 0.132, 0.167, 0.236, 0.236 and 0.317 in phase-LP, respectively. Interestingly, the strongest Tc–O₅ bond is about 2.5 times as much as the weakest Tc–O₁ bond. Such inhomogeneous chemical bonding character indicates huge distortion of the TcO₅ hexahedrons in phase-LP. Comparing to the relatively homogeneous Tc–O bonding character in phase-AC, these inhomogeneous Tc–O bonds result in a stronger anisotropy in phase-LP, which will be evidenced by the following analysis on the sound speed of TcO₂.

Fig. 4 The electron localization function (ELF) profiles for the Tc–O bonds in (a) distorted TcO₆ octahedrons of phase-AC and (b) distorted TcO₅ hexahedrons of phase-LP.

To further study the structure homogeneity of the polyhedrons, we analysis the O–Tc–O bond angle variance in the TcO₆ octahedrons and TcO₅ hexahedrons according to the following definition of bond angle deviation σ:

where n = 12 for TcO₆ octahedron and n = 9 for TcO₅ hexahedron. The calculated O–Tc–O bond angles θ_i in the TcO₆ octahedron in phase-AC and TcO₅ hexahedron in phase-LP are listed in Table 4, the ideal bond angles θ_i,ideal in octahedron and hexahedron are shown for comparison as well. According to Table 4, the calculated bond angle deviation σ_AC = 4.55° for the TcO₆ octahedron in phase-AC and σ_LP = 9.26° for the TcO₅ hexahedron in phase-LP, respectively. Hence the TcO₅ hexahedron in phase-LP is more distorted than the TcO₆ octahedron in phase-AC, which agrees well with our previously structure and chemical bonding analysis.

Table 4 The calculated O–Tc–O bond angle (°) of TcO₂ in the polyhedrons of different phases

O–Tc–O bond angle	Octahedrons		Hexahedrons
	TcO6	Ideal	TcO5	Ideal
θ1	98.70	90.00	127.91	120.00
θ2	93.78	90.00	122.23	120.00
θ3	93.46	90.00	106.81	120.00
θ4	92.40	90.00	103.40	90.00
θ5	91.33	90.00	98.33	90.00
θ6	90.92	90.00	90.93	90.00
θ7	89.66	90.00	90.34	90.00
θ8	89.25	90.00	87.19	90.00
θ9	87.05	90.00	73.50	90.00
θ10	86.70	90.00	—	—
θ11	85.19	90.00	—	—
θ12	80.27	90.00	—	—

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According to the first Brillouin zone (1BZ) and related high symmetry points for monoclinic crystal³⁴ in Fig. 1(b), we unfolded the electronic band structures for TcO₂ as well as the electronic density of states (DOS) in Fig. 5. Note that there are bands cross the Fermi level and show finite values at the Fermi level of the electronic DOS for both the phases, indicating that TcO₂ displays a metallic conductivity. Meanwhile, the Fermi level locates at the shoulder of the peaks of the total electronic DOS confirming the electronic stability for both the phases. For phase-AC, it is observed that the hybridization of 4d orbital of Tc the 2p orbital of O contributes to the Tc–O bonds. There shows a pseudo-gap around −1.25 eV and a small gap around −2.5 eV. The bands below −2.5 eV are mainly occupied by O 2p electrons, while the bands above are dominated by Tc 4d states. For phase-AC, the pseudo-gap opens at −1.5 eV. Beyond that, the overall features of the electronic DOS are similar for these two phases. However, phase-AC and phase-LP exhibit very different band structure character. First of all, the energy bands in phase-LP are more flat than that in phase-AC, showing heavier electron effective mass and indicating higher electrical resistivity for phase-LP. Secondly, there exhibits many double degenerated bands at Γ point in phase-AC. But because of the large structure distortion, no degeneracy bands at the Γ point can be found in phase-LP. Thirdly, the metallic nature of phase-AC mainly comes from the bands across the Fermi level around point in 1BZ as well as and point. About 1 eV gap at Γ point presents no contribution to its metallic behavior. For phase-LP, the bands around Z point do not cross with the Fermi level. Its metallic nature primarily comes from the band around X point. The bands around Γ point and Y point partially contribute to the metallic nature as well.

Fig. 5 The calculated band structure and density of states (DOS) of TcO₂ in (a) phase-AC at 0 GPa and (b) phase-LP under −13.3 GPa negative pressure.

To study the mechanical stability and lattice vibration properties of TcO₂ in phase-AC and phase-LP, we have calculated the phonon dispersion curves as well as the phonon DOS, illustrated in Fig. 6. As can be seen, no negative or imaginary frequency was found, suggesting that TcO₂ shows good lattice dynamical stability in phase-AC at ambient condition or in phase-LP under −13.3 GPa negative pressure. According to the phonon dispersion curves, obvious structural difference can be found. The maximum frequencies of the optical modes are ∼22.5 THz for phase-AC and ∼20.5 THz for phase-LP. The phonon dispersions show a huge frequency gap from 17 to 22 THz for phase-AC (from 15 to 19 THz for phase-LP) and a pseudo gap around 13 THz for phase-AC (10 THz for phase-LP). Moreover, the light O atoms vibrate in the high frequency region, whereas the heavy Tc atoms vibrate in the low frequency region for both the two phases. However, phase-AC and phase-LP show distinguished features in the intermediate frequency part, indicating the different Tc–O interactions. Phase-AC shows a phonon DOS valley around 7 THz, revealing weak resonance between Tc and O atoms. Meanwhile, strong sympathetic vibrations around 5–10 THz are observed in the phonon DOS for phase-AC. This is because the TcO₆-octahedrons in phase-AC are more regular than the TcO₅-hexahedrons in phase-LP. The sound velocities are further investigated by fitting the slopes of the acoustic dispersion curves around the Γ point. The maximum speed of longitudinal sound along the Γ–Z, Γ–X and Γ–Y directions are ν^Γ–Z_s = 6.69 km s⁻¹, ν^Γ–X_s = 7.65 km s⁻¹ and ν^Γ–Y_s = 7.18 km s⁻¹ for phase-AC, respectively. And the maximum sound velocity are ν^Γ–Z_s = 2.52 km s⁻¹, ν^Γ–X_s = 5.97 km s⁻¹ and ν^Γ–Y_s = 3.56 km s⁻¹ for phase-LP along the Γ–Z, Γ–X and Γ–Y directions, respectively. It is clearly shown that phase-LP exhibits stronger anisotropy than phase-AC, which is attributed to the more inhomogeneous Tc–O bonding in the low pressure phase as discussed above.

Fig. 6 The calculated phonon dispersion curves and phonon DOS of TcO₂ in (a) phase-AC at 0 GPa and (b) phase-LP under −13.3 GPa negative pressure.

To further analyze the stability of phase-LP, we plotted the phonon dispersion curves and the phonon DOS for phase-LP under ambient condition (0 GPa) in Fig. 7(a). The acoustic phonon mode shows a small imaginary part, indicating that the lattice is slightly unstable. This imaginary acoustic mode is enhanced to positive at no less than −3.7 GPa negative pressure (showing in Fig. 7(b)). Moreover, similar to the imaginary modes in ThH₂ (ref. 35) and Ge₂Sb₂Te₅,³⁶ such small instability can be easily stabilized by the temperature induced electron phonon interactions, which protects the phase stability of phase-LP under up to 5.1 GPa positive external pressure in Fig. 2(c).

Fig. 7 The calculated phonon dispersion curves and phonon DOS of TcO₂ in phase-LP (a) at ambient conditions and (b) under −3.7 GPa negative pressure.

4. Conclusion

In summary, we have systematically studied the crystal structure, chemical bonding, electronic structure and lattice dynamical properties for TcO₂. A stable low pressure phase (phase-LP) with different crystallographic features to the ambient condition phase (phase-AC) was predicted by ab initio random structure search. We found that the TcO₆-octahedrons in TcO₂ transform to TcO₅-hexahedrons during the phase transition from phase-AC to phase-LP. Large voids can be observed along the (001) direction in phase-LP only, which can be the transport channel for the protons and β particles. According to the chemical bonding and lattice dynamical analysis, the TcO₅-hexahedrons in phase-LP are more distorted and show stronger anisotropy than the TcO₆-octahedrons in phase-AC. The origins of the metallic nature of phase-AC and phase-LP are different as well. The stability of phase-LP is further verified by analyzing the lattice dynamical properties.

Acknowledgements

This work was supported by National Natural Science Foundation for Distinguished Young Scientists of China (51225205), the National Natural Science Foundation of China (61274005 and 51171046) and the Ph.D. Programs Foundation of Ministry of Education of China (no. 20133514110006).

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