Magnetic structure of (C₅H₁₂N)CuBr₃: origin of the uniform Heisenberg chain behavior and the magnetic anisotropy of the Cu²⁺ (S = 1/2) ions

Abstract

The magnetic properties and electric polarization of the organic/inorganic hybrid system (C₅H₁₂N)CuBr₃ (C₅H₁₂N = piperidinium) were examined on the basis of density functional theory calculations. The spin exchanges of (C₅H₁₂N)CuBr₃ evaluated by energy-mapping analysis show that its uniform Heisenberg antiferromagnetic chain behavior is not caused by the CuBr₃ chains made up of edge-sharing CuBr₅ square pyramids, but by the two-leg spin ladders resulting from interchain interactions. The magnetic anisotropy of the Cu²⁺ ions in (C₅H₁₂N)CuBr₃ originates largely from the Br⁻ ligands rather than the Cu²⁺ ions. The electric polarization of (C₅H₁₂N)CuBr₃ arises from the absence of inversion symmetry in the crystal structure, and is weakly affected by the magnetic structure.

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1. Introduction

Recently a metal organic complex (C₅H₁₂N)CuBr₃ (C₅H₁₂N = piperidinium) was synthesized, and its crystal and magnetic structures were investigated.¹ The structural building units of (C₅H₁₂N)CuBr₃, which crystallizes in a monoclinic space group C₂/c, are distorted CuBr₅ square pyramids containing Cu²⁺ (d⁹, S = 1/2) ions with three nonequivalent Br atoms (Fig. 1a) (precisely speaking, the four basal Br atoms of each CuBr₅ are not coplanar but form a butterfly shape with trans ∠Br–Cu–Br = 151.50° and 177.95°). The CuBr₅ pyramids share their basal edges to form Cu₂Br₈ dimers (Fig. 1a), which in turn share their non-basal edges to form CuBr₃ chains (Fig. 1b) running along the c-direction, and these chains are surrounded by the C₅H₁₂N cations (Fig. 2). (C₅H₁₂N)CuBr₃ has no inversion symmetry in crystal structure, so it would be polar regardless of whether its Cu²⁺ spins undergo a magnetic ordering or not. However, it is of interest to examine how strongly the electric polarization of (C₅H₁₂N)CuBr₃ is influenced by its magnetic structure.

Fig. 1 Structural features of (C₅H₁₂N)CuBr₃, where the Cu and Br atoms are represented by blue and grey spheres, respectively (a) Cu₂Br₈ dimer unit made up of two CuBr₅ square pyramids by sharing their basal edges, where the numbers 1, 2 and 3 refer to Br(1), Br(2) and Br(3), respectively. (b) CuBr₃ chain obtained from Cu₂Br₈ dimers by sharing their non-basal edges. (c) Arrangements of Cu₂Br₈ dimer units in the bc*-plane, where the apical Br atoms are removed for clarity. (d) Arrangement of two CuBr₃ chains. The numbers 1–6 in (b)–(d) refer to the spin exchange paths J₁–J₆, respectively.

Fig. 2 Two views of how the CuBr₃ chains are packed with C₅H₁₂N⁺ cations in (C₅H₁₂N)CuBr₃: projection views (a) along the c-direction and (b) along the b-direction. The pink polyhedra, black, gray, and blue circles are represent the CuBr₅ units, H, C, and C atoms, respectively.

The temperature-dependent magnetic susceptibility measured for powder samples of (C₅H₁₂N)CuBr₃ is well described by a Heisenberg uniform antiferromagnetic (AFM) chain model down to 1.8 K, and (C₅H₁₂N)CuBr₃ undergoes a three-dimensional (3D) magnetic ordering below T_N = 1.68 K.¹ The magnetic orbital of each CuBr₅ square pyramid is the x² − y² orbital lying in the basal plane of the pyramid (Fig. 3a). Thus, within each CuBr₃ chain, there are two different nearest-neighbor spin exchanges. In the exchange path J₁, the two magnetic orbitals can interact strongly because they are coplanar (Fig. 1b). In the exchange path J₂, the two magnetic orbitals cannot interact strongly because they are not coplanar (Fig. 1b). Consequently, the spin exchanges J₁ and J₂ cannot be identical so that the CuBr₃ chains cannot be responsible for the uniform AFM chain behavior observed in experiments. A chosen spin–lattice such as the Heisenberg uniform AFM chain model should be consistent with its the electronic structure, which determines the magnetic energy spectrum.^2,3 Experimentally, the spin-exchange parameters of a chosen spin–lattice are determined as the fitting parameters that simulate well the experimental magnetic data. However, the correctness of a chosen spin–lattice is not necessarily guaranteed even if it provides a good fitting as found for (VO)₂P₂O₇,^4,5 Na₃Cu₂SbO₆ and Na₂Cu₂TeO₆,^6–10 Bi₄Cu₃V₂O₁₄,^11–14 and Cu₃(CO₃)₂(OH)₂,^15,16 to name a few. To find what spin exchange paths of (C₅H₁₂N)CuBr₃ are responsible for its uniform antiferromagnetic AFM chain behavior, it is necessary to evaluate the intrachain as well as the interchain spin exchanges (Fig. 1).

Fig. 3 (a) The x² − y² state of a Cu²⁺ ion at a square planar, a square pyramidal or an octahedral site. (b) Arrangement of two x² − y² magnetic orbitals leading to a strong antiferromagnetic interaction. (c) The xy state of a Cu²⁺ ion at a square planar, a square pyramidal or an octahedral site.

Fig. 4 Temperature-dependent magnetic susceptibility of (C₅H₁₂N)CuBr₃, χ, calculated by using the Monte Carlo method on the basis of the classical spin Hamiltonian defined in terms of the spin exchange constants obtained from DFT+U calculations (U^eff = 6.0 eV). The inset shows the experimental magnetic susceptibility taken from ref. 1.

Another interesting magnetic property of (C₅H₁₂N)CuBr₃ is its magnetic anisotropy. The magnetic susceptibilities measured for single-crystal samples of (C₅H₁₂N)CuBr₃ with probe magnetic field applied along the b-, c- and a*-directions¹ show that the susceptibility is substantially stronger along the a*-direction than along the b- and c-directions. As depicted in Fig. 1c, the basal planes of the CuBr₅ square pyramids are approximately parallel to the bc*-plane. Thus, the preferred spin orientation of the Cu²⁺ ions in (C₅H₁₂N)CuBr₃ is expected to be perpendicular to the basal plane (easy-axis anisotropy), i.e., along the (a–c/2)-direction, although this direction was not probed experimentally. Many magnetic solids containing Cu²⁺ ions show typically the easy-plane anisotropy as found for CuCl₂·2H₂O,^17,18 CuCl₂,^19,20 CuBr₂,²¹ LiCuVO₄ (ref. 22) and Bi₂CuO₄.^18,23,24 For nearly six decades, it had been erroneously believed that spin-1/2 ions embedded in solids cannot have magnetic anisotropy arising from spin–orbit coupling (SOC), so their magnetic anisotropy is caused either by anisotropic spin exchange or by their magnetic dipole–dipole interactions.²⁵ It is true that SOC cannot generate magnetic anisotropy for isolated spin-1/2 ions. However, it is recently reported that the spin-1/2 ions embedded in solids do possess the SOC-driven magnetic anisotropy because the d-states of such ions are split by the crystal field of their surrounding ligands.¹⁸ The easy-axis anisotropy, which the Cu²⁺ ions in (C₅H₁₂N)CuBr₃ appear to exhibit, is found for Li₂CuO₂.^26,27 Since their magnetic anisotropy is expected to be caused by the SOC-induced interactions between the crystal-field split d-states,^3,18 we may speculate that the split d-states of the Cu²⁺ ions in (C₅H₁₂N)CuBr₃ differ from those of the Cu²⁺ ions in compounds showing in-plane anisotropy. Furthermore, since the SOC constant is greater for Br than for Cu by a factor of ∼2.9,²⁸ we may also speculate that the Br ligands have an important role in the spin orientation of Cu²⁺ ion.

In the present work we explore the three questions raised above on the basis of density functional theory (DFT) calculations. We evaluate the intrachain and interchain spin exchanges (J₁–J₆) of (C₅H₁₂N)CuBr₃ by performing energy-mapping analysis to find that two-leg-spin ladders with strong AFM rung act effectively as uniform AFM chains. Our DFT calculations show that the easy-axis anisotropy of Cu²⁺ ions of (C₅H₁₂N)CuBr₃ is largely induced by the SOC of Br⁻ ligands rather than that of Cu²⁺. Finally we show that the electric polarization of (C₅H₁₂N)CuBr₃ is not much affected by the change in its magnetic structure.

2. Computational details

In our DFT calculations, we employed the frozen-core projector augmented wave method^29,30 encoded in the Vienna ab initio simulation package (VASP),³¹ and the generalized-gradient approximation of Perdew, Burke and Ernzerhof³² for the exchange–correlation functional with the plane-wave-cut-off energy of 450 eV and a set of 32 k-point for the irreducible Brillouin zone. To examine the effect of the electron correlation in the Cu 3d states, the DFT plus on-site repulsion method (DFT+U)³³ was used with the effective U^eff = U − J values of 2, 4, 6 and 8 eV. The preferred orientation of the Cu²⁺ spins in (C₅H₁₂N)CuBr₃ was determined by performing DFT+U calculations including the SOC.³⁴

3. Spin exchange and spin lattice

Spin exchanges in magnetic solids of spin-1/2 Cu²⁺ ions are strongly governed by the arrangement of the square planes containing their magnetic orbitals.^2,3 J₁ and J₂ are of the Cu–Br–Cu exchange type, while J₃–J₆ are of the Cu–Br⋯Br–Cu exchange type. J₁, J₂, and J₃ are intrachain exchanges, and the J₄, J₅, and J₆ are interchain exchanges. The geometrical arrangements of the exchange paths J₁–J₆ is presented in Fig. 5. The geometrical parameters associated with the paths J₁–J₆ are summarized in Table 1. In terms of the projected density of states (PDOS) plots the electronic structure calculated for the ferromagnetic (FM) state of (C₅H₁₂N)CuBr₃ is presented in Fig. 6. The PDOS plots of the Cu 3d and Br 4p orbitals, shown only for the down-spin (minority-spin) states for simplicity, are obtained by using the local Cartesian coordinated defined in Fig. 6a. The contribution to the unoccupied states come primarily from the x² − y² orbital of Cu, the 4p_x orbitals of Br(1), 4p_y orbitals of Br(3), as well as the 4p_x/4p_y orbitals of Br(2). Each Br(2) has both 4p_x and 4p_y orbital contributions, because they make σ* bonding combinations to the x² − y² orbitals to the two different Cu atoms (Fig. 6a). This is consistent with the nature of the magnetic orbital expected for the Cu²⁺ (S = 1/2, d⁹) ion of each CuBr₅ square pyramid (Fig. 3a).

Fig. 5 Arrangements of the spin exchange paths in (C₅H₁₂N)CuBr₃: (a) a layer made up of the exchange paths J₁–J₅. (b) Arrangement of the exchanges J₁–J₆ forming a 3D lattice.

Table 1 Geometrical parameters associated with the exchange paths J₁–J₆ in (C₅H₁₂N)CuBr₃. The lengths and angles are in units of Å and degrees, respectively

	Cu–Cu	Cu–Br	Br⋯Br	∠Cu–Br–Cu	∠Cu–Br–Br
J1	3.622	2.462 (×2)		95.0
		2.449 (×2)
J2	3.712	2.423 (×2)		90.2
		2.802 (×2)
J3	6.673	2.423	4.009		100.5
		2.462			93.9
J4	7.348	2.412 (×2)	4.220 (×2)		104.0
		2.463 (×2)	3.929		161.9
					113.8
J5	8.835	2.449 (×2)	4.220		154.6
					161.9
J6	9.357	2.412	6.560		119.5
		2.423			117.6

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Fig. 6 PDOS plots obtained for the down-spin d-states in the FM state of (C₅H₁₂N)CuBr₃ from the DFT+U calculations with U^eff = 4 V: (a) the local Cartesian coordinate used to obtain the PDOS plots. (b) The PDOS plots obtained for the Cu 3d orbitals. (c) The PDOS plots obtained for the Br 4p orbitals.

To extract the values of J₁–J₆ by energy-mapping analysis, we first calculate the six relative energies using the seven ordered spin states (FM, AF1–AF6) of (C₅H₁₂N)CuBr₃, depicted in Fig. S1,† on the basis of DFT+U calculations as summarized in the parentheses of Fig. S1.† In terms of the spin Hamiltonian

where J_ij = J₁–J₆, the total spin exchange energies E_spin of these states per eight formula units (8FUs) are expressed as

E_spin = (n₁J₁ + n₂J₂ + n₃J₃ + n₄J₄ + n₅J₅ + n₆J₆)N²/4 (2)

by using the energy expressions obtained for spin dimers with N unpaired spins per spin site (in the present case, N = 1).^3,35 The values of n₁–n₇ for the seven ordered spin states, FM and AF1–AF6, are summarized in Table S1.† Thus, by mapping the relative energies of the spin ordered states obtained from the DFT+U calculations onto the corresponding relative energies from the total spin exchange energies, we obtain the values of J₁–J₆ summarized in Table 2.

Table 2 Values of the spin exchanges J₁–J₆ (in k_BK) of (C₅H₁₂N)CuBr₃ obtained from the DFT+U calculations with effective U^eff (in eV)

	U^eff = 2	U^eff = 4	U^eff = 6	U^eff = 8
J1	−197.6	−105.9	−52.3	−21.4
J2	12.1	14.4	15.1	14.6
J3	−5.7	−4.1	−3.1	−2.5
J4	−9.5	−4.5	−4.5	−3.5
J5	−41.8	−33.7	−27.1	−22.6
J6	0.05	0.02	−0.04	−0.25

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Among the intrachain spin exchanges, J₁ is strongly AFM, J₂ is ferromagnetic (FM), and J₃ is negligible. Given that ∠Cu–Br–Cu = 95° and 90° for the J₁ and J₂ paths, respectively, it is not surprising that J₁ is AFM while J₂ is FM.³⁶ Clearly, then, the CuBr₃ chains do not form uniform AFM chains as anticipated. Among the interchain exchanges, J₄ and J₆ are weak, but J₅ is strongly AFM although the Cu⋯Cu distance is very long compared to the intrachain case. Note that uniform AFM chains should be formed along the b-direction with the J₅ exchange. In the Cu–Br⋯Br–Cu exchange J₅, the two magnetic orbitals are arranged as depicted in Fig. 3b. Thus, the overlap between the two magnetic orbitals across the Br⋯Br contact via the Br 4p magnetic orbital tails is strong, thereby leading to a strong AFM exchange. Note that two adjacent uniform chains made up of the interchain exchanges J₅ are linked by the intrachain exchanges J₁ to form two-leg spin ladders. Such ladders are linked by the FM intrachain exchanges J₂ to form 2D layers of the two-leg spin ladders parallel to the bc-plane, and these 2D layers are stacked along the a-direction (Fig. 5) with very weak interlayer exchange J₆ between them. J₆ is weakly FM for U^eff < 5 eV, but is weakly AFM for U^eff > 5 eV. In any event, the existence of a nonzero J₆ allows (C₅H₁₂N)CuBr₃ to undergo a 3D AFM ordering at low temperature as observed experimentally.¹

For our discussion of the two-leg spin ladders, it is convenient to employ the new notations J_‖ = J₅ for the leg and J_⊥ = J₁ for the rung. Table 2 shows that the J_⊥/J_‖ ratio changes from 4.7 to 0.95 as U^eff varies from 2 to 8 eV. When the rung exchange J_⊥ is considerably greater than the leg exchange J_‖, the two-leg spin ladder would behave like a uniform AFM chain, because the low-energy excitation spectrum would be dominated by the excitations associated with the weaker exchange (i.e., J_‖), not with the stronger exchange (i.e., J_⊥).

To verify this point, we simulate the magnetic susceptibility using the Monte Carlo method.^37,38 In this simulation we employed the values of the spin exchanges obtained from the DFT+U calculations with U^eff = 6 eV. As shown in Fig. 4, the calculated magnetic susceptibility is in good agreement with the experimental one. It is interesting to note that the uniform AFM chain is formed along the interchain direction instead of the intrachain direction.

4. Preferred spin orientation

To determine the preferred orientation of the Cu²⁺ spins in (C₅H₁₂N)CuBr₃, we carry out DFT+U calculations including the SOC for several different spin orientations defined with respect to the local Cartesian coordinate defined in Fig. 1a and 6a. There are eight identical Cu²⁺ ions per unit cell in (C₅H₁₂N)CuBr₃ but we simplify our calculations by replacing all but one Cu²⁺ ions with nonmagnetic Mg²⁺ ions. Results of our DFT+U+SOC calculations for this model are summarized in Table 3, which shows that the spin orientation along the local z-direction is lower in energy, though slightly, than that any other direction. That is, the Cu²⁺ (d⁹) ions in (C₅H₁₂N)CuBr₃ have easy-axis anisotropy, as already anticipated.

Table 3 Relative energies (in k_BK per Cu) of various orientations of the Cu²⁺ spin in (C₅H₁₂N)CuBr₃ obtained from the DFT+U+SOC calculations with effective U^eff (in eV). The spin orientation is defined using the local Cartesian coordinate (Fig. 1a and 5a)

Orientation	U^eff = 2	U^eff = 4	U^eff = 6
x	0.0	0.0	0.0
y	−0.3	−0.4	−0.5
z	−1.4	−1.7	−2.1
x + z	−0.5	−0.7	−0.9
y + z	−0.6	−0.8	−1.0

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We now examine how the above result can be explained from the viewpoint of the crystal-field split d-states of the CuBr₅ square pyramid. By employing the coordinate (x, y, z) for the orbital momentum and the coordinate (x′, y′, z′) for the spin momentum, the SOC term λŜ·L̂ is written as^3,39–41

where we omitted the terms that allow interactions between different spin states. The preferred spin orientation is given by the orientation of the z′-axis. If the z′-axis is along the z-axis (i.e., θ = 0°), the magnetic ion has easy-axis anisotropy. If the z′-axis lies in the xy-plane (i.e., θ = 90°), the magnetic ion has easy-plane anisotropy.

When an occupied up-spin (down-spin) d-state ψ_o↑ (ψ_o↓) of energy e_o interacts with an unoccupied up-spin (down-spin) d-state ψ_u↑ (ψ_u↓) of energy e_u via the matrix element 〈ψ_o|Ĥ⁰_SO|ψ_u〉, the associated energy lowering ΔE_SOC is given by

Provided that the matrix elements 〈ψ_o|Ĥ⁰_SO|ψ_u〉 are comparable in magnitude, the most important interaction is the one involving the highest occupied (HO) and the lowest unoccupied (LU) states. To predict the preferred spin orientation using eqn (3) and (4), it is necessary to know at what spin orientation the term 〈ψ_o|Ĥ⁰_SO|ψ_u〉 is nonzero and can be maximized. The preference for the ‖z direction (easy-axis anisotropy) requires a nonzero 〈ψ_o|L̂_z|ψ_u〉, while that for the ‖xy plane (easy-plane anisotropy) requires a nonzero 〈ψ_o|L̂₊|ψ_u〉 or a nonzero 〈ψ_o|L̂₋|ψ_u〉. In terms of the spherical harmonics Y₂^m (m = 0, ±1, ±2), the angular behaviors of the d-orbitals are given by 3z² − r² ∝ Y⁰₂, xz ∝ (Y₂⁻¹ − Y₂¹), yz ∝ (Y₂⁻¹ + Y₂¹), xy ∝ (Y₂⁻² − Y₂²), and x² − y² ∝ (Y₂⁻² + Y₂²). Namely, the difference in the magnetic quantum numbers m (i.e., |Δm|) is 0 between xz and yz and between xy and x² − y², |Δm| = 1 between 3z² − r² and {xz, yz} and between {xz, yz} and {xy, x² − y²}, and |Δm| = 2 between 3z² − r² and {xy, x² − y²}. Then, according to the relationship

the interaction between two d-states under the SOC induces easy-axis anisotropy if |Δm| = 0, but easy-plane anisotropy if |Δm| = 1.

Therefore, the easy-axis anisotropy found for the Cu²⁺ ions of (C₅H₁₂N)CuBr₃ is explained if the SOC-induced interaction of the empty x² − y²↓ states with the filled xy↓ states is stronger than that with the filled xz/yz↓ states. The PDOS plots of (C₅H₁₂N)CuBr₃ in Fig. 6b show that the empty x² − y²↓ orbitals are well separated from the filled xz/yz↓ orbitals, and equally well separated from the filled xy↓ orbitals. Using eqn (4) and the PDOS plots of Fig. 6b, however, it is difficult to conclude that the preferred spin orientation is the local z-direction (i.e., easy-axis anisotropy). The latter implies that the SOC of Cu is not responsible for the easy-axis anisotropy. Since the SOC constant of Br is about three times greater than that of Cu, one might wonder if the SOC of Br plays a role in determining the prefered spin orientation of (C₅H₁₂N)CuBr₃. Thus we examine the Br 4p orbitals combined into the xy↓ and x² − y²↓ states (Fig. 6c). The Br 4p orbitals make π* antibonding to the Cu xy orbital in the xy↓ state (Fig. 3c), but make σ* antibonding to the Cu x² − y² orbitals in the x² − y²↓ states (Fig. 3a). Thus, at any Br site of the CuBr₅ basal plane, the Br 4p orbital of the xy↓ is orthogonal to that of the x² − y²↓ state. At a given Br atom, these Br 4p orbitals can be taken to be 4p_x and 4p_y orbitals without loss of generality. As can be seen from Fig. 6c, the energy gap between the occupied and unoccupied Br 4p orbitals is much smaller than that between the occupied and unoccupied Cu d-orbitals. Thus, according to eqn (4), the SOC effect of Br is more important than that of Cu. This conclusion is further reinforced by the facts that the SOC constant is much greater for Br than for Cu, and that there are more Br than Cu atoms in (C₅H₁₂N)CuBr₃.

The only nonzero matrix element for the SOC between the 4p_x and 4p_y orbitals occurs for L̂_z, namely, 〈y|L̂_z|x〉 = i.⁴⁰ Because this term comes with the factor cos θ (eqn (3)), it is maximized when θ = 0°. Therefore, the Br ligands associated with the SOC between the x² − y²↓ and xy↓ states predict easy-axis anisotropy for (C₅H₁₂N)CuBr₃. Obviously, from the SOC of the x² − y²↓ states with the xz↓ or yz↓, one predicts in-plane anisotropy by considering the SOC of the associated Br 4p orbitals. This explains why the magnetic anisotropy is weak in (C₅H₁₂N)CuBr₃.

5. Polarization

Since (C₅H₁₂N)CuBr₃ has the crystal structure with no inversion symmetry, it should have a nonzero electric polarization. We calculate the electric polarization of (C₅H₁₂N)CuBr₃ for the magnetic ground state (i.e., the AF4 state) as well as hypothetical FM state by using the Berry phase method^42,43 encoded in the VASP. For the electric polarization of (C₅H₁₂N)CuBr₃, our DFT+U+SOC calculations give 0.17 and 0.19 μC m⁻² for FM and AFM states, respectively. Thus, the electric polarization of (C₅H₁₂N)CuBr₃ is weakly affected by its magnetic structure.

6. Concluding remarks

Our study shows that the uniform Heisenberg antiferromagnetic chain behavior of (C₅H₁₂N)CuBr₃ is not caused by the CuBr₃ chains, but by the two-leg spin ladders resulting from interchain interactions. The Cu²⁺ ions in (C₅H₁₂N)CuBr₃ have easy-axis anisotropy, which arises largely from the Br⁻ ligands rather than the Cu²⁺ ions. (C₅H₁₂N)CuBr₃ has a nonzero electric polarization, which is weakly affected by its magnetic structure.

Acknowledgements

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2013R1A1A2060341, 2013R1A1A2006416) and by the resource of Supercomputing Center/Korea Institute of Science and Technology Information with supercomputing resources including technical support (KSC-2014-C1-52). Erjun Kan thanks NSFC (21203096, 51522206), NSF of Jiangsu Province (BK20130031), PAPD, the Fundamental Research Funds for the Central Universities (No. 30915011203), and the Shanghai Supercomputer Centre. This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

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Footnote

† Electronic supplementary information (ESI) available: Table S1 as well as Fig. S1 are available. See DOI: 10.1039/c5ra26341g

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