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Ferromagnetism in Monatomic Chains: SpinDependent Bandwidth Narrowing/Broadening Piyush Dua, Geunsik Lee, and Kwang S. Kim J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b06327 • Publication Date (Web): 31 Aug 2017 Downloaded from http://pubs.acs.org on September 1, 2017

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The Journal of Physical Chemistry

Ferromagnetism in Monatomic Chains: Spindependent Bandwidth Narrowing/Broadening Piyush Dua,*,†,‡ Geunsik Lee,*,† and Kwang S. Kim*,† †

Center for Superfunctional Materials, Department of Chemistry, Ulsan National Institute of Science and Technology (UNIST), 50 UNIST-gil, Ulsan, Korea ‡

Department of Physics, University of Petroleum and Energy Studies, Dehradun, Uttarakhand, 248007 India *Corresponding authors: [email protected] (ksk), [email protected] (gl), [email protected] (pd) ABSTRACT: We have analyzed the role of band narrowing/broadening in the magnetic and electronic properties of low dimensional itinerant strongly correlated electronic systems, which is particularly important in 3d transition elements. Density functional theory and mean field results have been coupled to explain the magnetic phenomenon in one dimensional monatomic chain of 3d transition elements Sc, Ti, V, Cr, Mn, Fe, Co and Ni. We found that in ferromagnetic ground state not only the band splitting but also the relative band narrowing/broadening plays a crucial role and the bandwidths of both spins become different as a manifestation of correlated spin hopping interaction. For less than half-filled band (i.e. for Sc, Ti, V and Cr), the up spin band broadens and the down spin band narrows down. For the (half and) more than half filled band (i.e. for Mn, Fe, Co and Ni), the down spin band broadens and the up spin band narrows down. As a result, (in most of the cases) only one spin channel is present for conduction, showing ferromagnetism which could be useful for spintronics.

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Introduction There have been extensive efforts to design and develop spintronic devices experimentally1-5 and theoretically6-10 because of their importance in information processing. Thus, understanding the electronic and magnetic properties of low dimensional systems and their transport phenomena is highly essential. The magnetic properties of 3 dimensional (3D) systems of itinerant electron systems are well understood. A number of studies have been carried out on magnetism and spin transport in low dimensional systems.11-13 The functionality of conventional nanodevices can be enhanced by using spin oriented transport.14-16 One dimensional (1D) and two dimensional (2D) electronic systems are expected to show unique behavior of their magnetic and electronic properties due to highly decreased density of states as compared with the 3D systems.17-28 Subsequently, the atomic nature and Coulomb/exchange interactions play important roles in spintronic device properties. However, the phenomenon of onset of magnetization in low dimensional systems is not yet clear. There are many questions unanswered about the magnetic and electronic properties of low dimensional itinerant electron systems. The magnetic and transport properties of 1D and 2D metal systems have been intensively studied.29-30 Jepsen et al.31 revealed that Ni surfaces and thin films show larger magnetism than the bulk and emphasized that band-narrowing is the effect of reduced coordination number. From the X-ray photoemission spectroscopy of Ni, Liebsch reported the band narrowing phenomenon while the band-splitting is less than expectation.32 Luo et al. found the half metallic behavior from the magnetism of 3-d transition metal nanowires (NWs) on wurtzite-boron nitride (0001) surfaces,33 while Chen et al. reported the spin polarization transport from the magnetism of Fe nanowires encapsulated inside the semiconducting inside silicon carbide nanotubes.34 By altering the geometric structures of metallic chains, Gao et al. tuned the electronic excitations.35 Syromyatnikov et al. demonstrated the existence of two nearly isoenergetic ferromagnetic states of Co wires on a vicinal Cu{111} surface.36 The contribution of exchange interactions in 3-d transition metal chains was investigated by Mokrousov et al.37-38 Yet, despite numerous studies on 1D ferromagnetic systems and spintronic nanodevices, there are questions to be cleared up; (i) what is responsible for the onset of magnetization in low dimensional systems? Is it exchange splitting, band narrowing/broadening, or their combined effect? (ii) Why is only one channel (either up or down) present for conduction in the ferromagnetic state upon lowering the dimension for metals?17,38-40 It has been well established since the inclusion of Coulomb/exchange correlations in the Hubbard model Hamiltonian helps obtain the correct ground state of the itinerant ferromagnets. Here we emphasize the effect of off-diagonal matrix elements of Coulomb/exchange interaction in low-dimensional itinerant ferromagnets. We assume that in going from 3D to 2D to 1D, the change in off-diagonal elements of Coulomb/exchange interaction is important because the intra-site Coulomb/exchange interaction does not change much. In this regard, 1D NWs of transition elements constitute the fundamental species to understand the physics of magnetism and conductance at low dimensions. These NWs show enhanced magnetism,


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as compared to their bulk (3D) state, and quantized conductance. They are predicted to be technologically useful for spintronics based devices.13,40 Various studies have been carried out on 1D monatomic chains41-42 including transition elements of 3-d series,17,38,43-55 4-d series17,37 5-d series17,30,56,57 and alkali metals.57-59 It was found that Pt nanowires exhibit Hund’s rule magnetism and the magnetic moment increases upon stretching, while in bulk state Pt does not show magnetism.56 Delin et al60 explored the spin dynamics of atomically thin platinum wires based on first-principles calculations. On the experimental side, it was demonstrated that transition atoms can form freely suspended chains using controllable break junctions.61 Gamberdella et al.2 demonstrated the first controlled realization of 1D monatomic metal chains, which shows enhanced magnetism and band narrowing of Co 3-d band with tunable anisotropy energy.62 From these results it can be pointed out that in going from bulk (3D) to linear chain (1D), the band narrowing is manifested because of less number of nearest neighbors. A reduction of the coordination number means that the bandwidth or the kinetic energy of the electron is reduced and the electrons have less opportunity to hop from site to site. Then, the ratio (U/W) of the Coulomb interaction U between the electrons on a given site to the bandwidth (W) increases, and so the electron correlations become more important and the tendencies towards the appearance of magnetism or a Mott-transition are enhanced. Another aspect which can be pointed out from these results is that in 1D chains of transition elements, (in most of the cases) the bandwidths of up and down spin bands are not equal, and only one spin channel is present at the Fermi level for conduction.17,37 Therefore, as far as bandwidths of up and down spins are concerned, two effects can be seen; (i) band-narrowing of both up and down spin bands as a manifestation of the reduced coordination number in going from 3D to 1D and (ii) a relative band narrowing/broadening of up and down spin bands at the onset of ferromagnetic state in 1D. The reason for the first aspect is quite clear as stated above but the second aspect has not received considerable attention. In the present work, we make an attempt to explain this aspect with the help of extended Hubbard model. Using extended Hubbard model, Kollar et al.63 studied the effect of correlated hopping using the Gutzwiller wavefunction to demonstrate spin dependent band-narrowing/broadening on half filling, as well as away from half filling. Despite a number of Hubbard model studies on ferromagnetic NWs,64-71 such a Hubbard model has not been well tested by using DFT. Thus, addressing the techniques used for qualitative derivation of variation of bandwidth with magnetization based on the Hubbard model in Supporting information, we report a quantitative description of results obtained from DFT calculations. Then, we attempt to correlate these qualitative and quantitative results, followed by concluding remarks. Calculation Methods


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To explain the bandwidth variation qualitatively, we have considered 1D extended Hubbard model, which includes all the off-diagonal matrix elements of Coulomb interaction. We have applied the Green’s function equation of motion approach to obtain quasi-particle energy. Within the mean field approximation it is found that at the onset of magnetization, the probability of hopping of an electron is spin density dependent. For quantitative study, we analyzed the bandwidth change and magnetic properties of 1D chains of 3-d transition elements. We have used the full-potential linear augmented plane-wave method as implemented in the WIEN2k72 code. The exchange and correlation effects were treated within DFT using the generalized gradient approximation (GGA).73 The tetrahedron74 and Gaussian smearing methods were used for Brillouin-zone integration for density of states and the Fermi energy (EF) calculation, respectively. We chose the plane wave cutoff (R * Kmax = 8 where R in a.u. is the smallest atomic sphere radius and Kmax in Ry1/2 is the largest K-vector in the plane wave expansion of the wave function). The separation between 1D monatomic infinite chains has been taken as 10 Å to avoid the interchain interaction. We have investigated the relativistic effects through spin-orbit coupling and found that the relativistic effects do not change the magnetic moment but there is a small difference in ground state energy. These calculations will be referred to as fully relativistic (FR) whose values are compared with the scalar relativistic (SR) values and the non-magnetic (NM) values. Core states have been treated fully relativistically in all the calculations. We analyze these results with the help of interaction parameters of the Hubbard model which includes diagonal and off-diagonal elements of Coulomb interaction.75 Hubbard model has been used to study the itinerant ferromagnets for quite long. It was investigated by Hirsch67 and thereafter Dua et al.66,68 that the inclusion of off-diagonal elements of Coulomb interaction in the Hamiltonian introduces band-narrowing/broadening, and it is the key concept for a system to become ferromagnetic. The Hamiltonian, which includes all the off-diagonal matrix elements of Coulomb interaction, is given by † † † 𝐻𝐻 = −t ∑〈𝑖𝑖𝑖𝑖〉𝜎𝜎�𝑐𝑐𝑖𝑖𝑖𝑖 𝑐𝑐𝑗𝑗𝑗𝑗 + h. c. � + U ∑𝑖𝑖 𝑛𝑛𝑖𝑖↑ 𝑛𝑛𝑖𝑖↓ + V ∑〈𝑖𝑖𝑖𝑖〉 𝑛𝑛𝑖𝑖 𝑛𝑛𝑗𝑗 + J ∑〈𝑖𝑖𝑖𝑖〉𝜎𝜎𝜎𝜎′ 𝑐𝑐𝑖𝑖𝑖𝑖 𝑐𝑐𝑗𝑗𝜎𝜎′ 𝑐𝑐𝑖𝑖𝜎𝜎′ 𝑐𝑐𝑗𝑗𝑗𝑗 + † † † † † P ∑〈𝑖𝑖𝑖𝑖〉�𝑐𝑐𝑖𝑖↑ 𝑐𝑐𝑖𝑖↓ 𝑐𝑐𝑗𝑗↓ 𝑐𝑐𝑗𝑗↑ + h. c. � + K ∑〈𝑖𝑖𝑖𝑖〉𝜎𝜎𝜎𝜎′ �𝑐𝑐𝑖𝑖𝑖𝑖 𝑐𝑐𝑖𝑖𝜎𝜎′ 𝑐𝑐𝑗𝑗𝜎𝜎′ 𝑐𝑐𝑖𝑖𝑖𝑖 + h. c. � − µ ∑𝑖𝑖𝑖𝑖�𝑐𝑐𝑖𝑖𝑖𝑖 𝑐𝑐𝑖𝑖𝑖𝑖 �

(1)

† In eq. 1 h.c. denotes Hermitian conjugate, and 𝑐𝑐𝑖𝑖𝑖𝑖 /𝑐𝑐𝑖𝑖𝑖𝑖 creates/destroys an electron of spin σ at site ‘i’. 𝑛𝑛𝑖𝑖𝑖𝑖 is the occupation number at the site ‘i’ with spin σ, and 𝑛𝑛𝑖𝑖 = 𝑛𝑛𝑖𝑖↑ + 𝑛𝑛𝑖𝑖↓ is the number of

electrons at site ‘i’. ‘t’ is the hopping parameter between sites ‘i’ and ‘j’, where 〈𝑖𝑖𝑖𝑖〉 indicates that

‘i’ and ‘j’ are nearest neighbors. U is the intra-site Coulomb repulsion. V, J, P and K are offdiagonal matrix elements representing inter-site Coulomb repulsion, exchange interaction, pairhopping interaction, and correlated-hopping interaction, respectively. μ is the chemical potential.


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In transition elements, d-electrons exhibit both characteristics of the ordinary band model and the atomic model. d-electrons can hop from site to another, therefore contribute to the conduction and exhibit magnetism as well. The Hubbard model takes into account both contributions and therefore first term in eq. 1 is the hopping term. It represents the kinetic energy of electrons hopping between atoms. The second term in diagonal form in the Hubbard model is then the on-site repulsion which represents the potential energy arising from the charges on the electrons on the same atom. All other terms except for the last chemical potential term are representing the potential energy from the charges on the nearest neighbor atom.

To obtain the quasi-particle energy, we use the Green’s function method76 with mapping the Coulomb interactions into the perturbation to the non-interacting electron kinetic energy 𝜀𝜀𝑘𝑘 = −t ∑𝛿𝛿 𝑒𝑒 −𝑖𝑖𝑖𝑖𝑖𝑖 (see Supplementary for details). The energy of the quasi-particle for non-degenerate extended Hubbard model within the mean field approximation is66,67 𝐸𝐸𝜎𝜎 (𝜀𝜀𝑘𝑘 ) = �1 −

2zIσ W

(J − V) −

2zI−σ W

(J + P) −

2zn−σ W

M

K� 𝜀𝜀𝑘𝑘 + �−σ 2 (U + zJ) + zKI−σ � − µ,

(2)

where M = n↓‒ n↑ is the magnetization per site, W = 2zt is the bandwidth and ‘z’ is the number of nearest neighbor. In the absence of off-diagonal elements of Coulomb interaction (i.e. when V, P, J, and K vanish), only band-splitting is present, while the inclusion of off-diagonal elements introduces band-narrowing/broadening in addition to band-splitting. The first part (within the bracket) i.e. the coefficient of 𝜀𝜀𝑘𝑘 in eq. (2) decides the band narrowing/broadening due to interactions, in which the role of Iσ is important and the second part is the band splitting part. For constant density of states, Iσ reduces to Iσ = nσ (1 − nσ ), which is dependent on the band-filling † (see Supplementary for details). Here nσ is different from 𝑛𝑛𝑖𝑖𝑖𝑖 = 𝑐𝑐𝑖𝑖𝑖𝑖 𝑐𝑐𝑖𝑖𝑖𝑖 that appears in Eq. (1). It is the averaged number of electrons per site within the mean-field approximation with either up or down spin. The average number of electrons occupied in the band is

= n n↑ + n↓

(3)

where 0 ≤ n ≤ 2 for a non-degenerate Hamiltonian. For less than half filled band, the conditions are (see Supplementary for details) n ≤n ≤n 2 ↑ ⇒ I↑ > I↓

and

n − n↑ ≤ n↓ ≤

n 2

(4)

for M ≠ 0

while for more than half filled band the conditions are (see Supplementary for details) n ≤ n ≤1 2 ↑ ⇒ I↓ > I↑

and

n − n↑ ≤ n↓ ≤

n 2

(5)

for M ≠ 0


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The effective mass ratio is defined as the inverse of bandwidth ratio (m*/m = W/W*),66,67 where W* is the bandwidth after inclusion of off-diagonal elements. In the paramagnetic region, the effective mass ratio is same for both spins. As the onset of ferromagnetism takes place under suitable circumstances, the effective mass ratio for down spin electrons decreases and that for up spin electrons increases. It is one of the primary requirements for spintronic devices that in ferromagnetic state the resistance must be less for one type of spin than for the other type. The change of effective mass ratio in the ferromagnetic state is driven by the off-diagonal matrix elements. Results and Discussion For systems with less than half-filled band, the probability of hopping for both spins is equal in nonmagnetic state, while at the onset of ferromagnetic state, the probability of hopping for spin up electron is greater than that for down spin. On the other hand, for the systems with more than half filled band, the probability of hopping for down spin electron is enhanced at the onset of ferromagnetic state. In 3-d itinerant electron systems, the inter-site correlated hopping (K) is much stronger than the inter-site exchange interaction (J),77 because the typical order of energy of Coulomb interactions in transition elements is U ~ 10 eV, V ~ 2-3 eV, J (and P) ~ .025 eV and K ~ 0.5 eV. Using the specified range of parameters, we have plotted in Figures 1(a-d) the bandwidth ratio as a function of normalized magnetization for different band fillings i.e. for n = 0.2, 0.4, 0.6 and 0.8, respectively. Two sets of Hamiltonian parameters have been used (i) J/W = j = 0.1, K/W = k = 0.5, V/W = v = 3.0 and (ii) j = 0.02, k = 0.5, v = 3.0. There is no significant change in the qualitative behavior of plots on changing the value of j from 0.1 to 0.02. From these Figures, it can be seen that for less than half filled band, the bandwidth of up and down spin is same in the absence of magnetization and at the onset of magnetization, the change in bandwidth is different for both spins. The bandwidth of up spin remains more than the bandwidth of down spin for finite value of magnetism. The effect is enhanced if the band-filling is increased.


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3

3

(a) n = 0.2

(e) n = 1.0

2

2

1

1

0 3

0 3

(b) n = 0.4

(f) n = 1.2

2

2

1

1

0 3

0 3

(c) n = 0.6

(g) n = 1.4

2

2

1

1

0 3

0 3

(d) n = 0.8 2

W* / W

W* / W

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1

0

(h) n = 1.6

2

1

0 0.0

0.2

0.4

0.6

m / mmax

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

m / mmax

Figure 1. Bandwidth ratio (W*/W) as a function of (normalized) magnetization for (a-h) n = 0.21.6. Left column is for the less than half filled, and right is for (half and) the more than half filled band. Upper (↑) and lower (↓) arrows denote the bandwidth for the up and down spins, respectively. Two sets of Hamiltonian parameters have been used (i) j = 0.1, k = 0.5, v = 3.0 (blue color) and (ii) j = 0.02, k = 0.5, v = 3.0 (red color). The same information is plotted for (half and) more than half filled band systems in Figures 1(eh) for n = 1.0, 1.2, 1.4 and 1.6 which shows that the bandwidth of down spin is greater than that of up spin for non-zero magnetization. The well known band-splitting effect is not discussed here. For quantitative analysis, we have calculated the cohesive energy Ec as a function of nearest neighbor distance. The minimum of Ec gives the equilibrium bond-length. Our results for bond length are in good agreement with existing results.17,38,43,44 The magnetic moment and spin resolved density of states have been calculated at these bond length values presented in Table 1. Although energetically V, Cr and Mn favor the antiferromagnetic state, they become ferromagnetic on stretching.39 In Figures 2(a-d), we have plotted the spin resolved density of d-states at equilibrium bond length for 1D chain of Sc, Ti, V and Cr. In all the cases the band splitting is quite clear and in addition the bandwidths for up and down spin d-bands are not equal. Table 1: NM, SR and FR values of equilibrium bond-length (d in Å) and magnetic moment (M in µB) of 1D linear chains.


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System Sc Ti V Cr Mn Fe Co Ni

dNM 2.74 2.23 2.04 1.84 1.88 1.98 2.04 2.15

dSR 2.90 2.25 2.62 2.78 2.60 2.28 2.16 2.16

dFR 2.92 2.28 2.60 2.80 2.62 2.28 2.16 2.18

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MFR 1.81 0.83 4.00 5.12 4.63 3.23 2.14 1.23

The down spin band is getting narrower and is situated above EF, whereas the up spin band is broader than the down spin band and few up spin states are available at EF for conduction. In the cases of Sc and V, the magnetic moment in the cell is larger than the number of localized delectrons per atom.17,38,39 Certainly, it is a manifestation of s-d hybridization.38 For Ti, the magnetic moment is very small, hence band narrowing/broadening is not significant. For Mn, Fe, Co and Ni in Figures 2(e-h), it is clear that band narrowing/broadening is present in addition to band splitting. The up spin band is localized and situated well below EF and the down spin channel is present for conduction at EF.17 For such a system the up spin band is full, but only the down spin electron can hop from one to another site because of correlated hopping. Therefore the down spin channel is present at the Fermi level for conduction. From Figures 1(a-d), it is clear that for the case of less than half filled band, the band for down spin is narrower than the up spin band. Quantitative results also show the similar behavior for Sc, V and Cr. Ti is an exception because of small magnetic moment. As effective mass of a particle is inversely proportional to bandwidth, the effective mass ratio for up spin electrons becomes smaller than that for down spin electrons; the up spin electrons become the majority carriers in these systems. In the case of more than half filled band (Figures 1(e-h)), the band for down spin is broader than the up spin band. The same behavior is endorsed by Figures 2(e-h), for Mn, Fe, Co and Ni. The effective mass ratio becomes smaller for down spin electrons than for up spin electrons, and so the down spin electrons become the major carriers in these systems. Within DFT, the exchange correlation might be responsible for the bandwidth correction of both spins, but we believe that the inter-site Coulomb correlations, especially correlated hopping interactions, are responsible for the bandwidth correction as no significant change occurs in the qualitative behavior of the plots (Figure 1) on changing the inter-site exchange interaction from 0.1 to 0.02 eV. In the systems with less than half filled band, there is a possibility for both spins to be conductive. Because there are much more up spin electrons than down spin, up spin electrons are the majority charge carriers with a finite density of states at EF. The same conclusion can be


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deduced directly from eq. (4) that the probability of hopping is higher for up spin than for down spin. In the systems with more than half filled band, it is difficult for up spin electrons to hop from one site to another (because up spin band is full) and there are a sufficient number of down spin electrons which can contribute to the conduction. Therefore the down spin electrons are the major charge carriers as evident from eq. (5).

Figure 2. Spin resolved density of d-states for (a) Sc, (b) Ti, (c) V, (d) Cr, (e) Mn, (f) Fe, (g) Co and (h) Ni, respectively at equilibrium value of nearest neighbor distance. On energy scale ‘0’ denotes EF. ↑ and ↓ denotes the up and down spin, respectively.


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Interestingly our DFT calculations reveal that the band narrowing/broadening is suppressed for 4d and 5d elements. For Ru, Rh and Pd, in addition to band-splitting the up-spin band becomes narrower than the down-spin band, but the magnitude is smaller than that for 3d elements (Fe, Co and Ni). In the cases of Os, Ir and Pt, the band narrowing/broadening is very small because the band splitting is too small. Based on these results, it is clear that in these systems, the ferromagnetism is driven by the combined effect of band-splitting and band-narrowing/broadening. If the band-splitting is small as in the cases of 4-d and 5-d systems, it is difficult to see the bandnarrowing/broadening effect whereas the band-splitting is quite clear. Nevertheless, even the small band-narrowing/broadening effect plays a significant role in ferromagnetism.

Conclusions For the spintronic materials, two important requirements are that (i) the better system needs to have lower dimensions and (ii) only one spin channel should be available for conduction. 1D monatomic wires of 3-d series of periodic table fulfill both requirements, which can be seen from the quantitative results and explained by considering the importance of off-diagonal matrix elements of Coulomb interaction within the mean field theory. In the absence of off-diagonal matrix elements (i.e., no bandwidth correction emerges), the system is driven by usual band splitting phenomenon because the Coulomb correlations become extremely important. Relative bandwidth correction is witnessed from the DFT results and is explained only with the inclusion of off-diagonal matrix elements into standard Hubbard model. Hence we believe that to have a good mapping between DFT and mean field results, one must consider the electron correlations by including the off-diagonal matrix elements of Coulomb interaction. Evident from the present study, the correlated hopping plays an important role to understand the physics of low dimensional itinerant ferromagnets.

ASSOCIATED CONTENT Supporting Information. The Supporting Information is available free of charge on the ACS Publications website. Extended Hubbard model and Green’s function method calculations for the quasi-particle energy considering the Coulomb interactions.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]


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*E-mail: [email protected] *E-mail: [email protected] ORCID Geunsik Lee: orcid.org/0000-0002-2477-9990 Kwang S. Kim: http://orcid.org/0000-0002-6929-5359 Notes The authors declare no competing financial interest. ACKNOWLEDGMENT This work was supported by the Korea government (MSIP) (No. 2014M3A9D8034459). REFERENCES

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