Controlling Oxygen-Based Electrochemical Reactions through Spin

Article pubs.acs.org/JPCC

Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

Controlling Oxygen-Based Electrochemical Reactions through Spin Orientation Satadeep Bhattacharjee† and Seung-Cheol Lee*,†,‡ †

Indo-Korea Science and Technology Center (IKST), Bangalore, India Electronic Materials Research Center, Korea Institute of Science & Techology, Seoul, Republic of Korea

‡

ABSTRACT: The role of spin orientation on the reactivity of oxygen reduction reaction (ORR) intermediates (O, OH) on a ferromagnetic electrode surface is studied using constrained density functional theory formalism. We show that the strength of the binding of these reaction intermediates depend on their relative spin orientations with respect to the magnetization of the electrode. This suggests that oxygen-based electrochemical reactions on ferromagnetic catalyst surfaces can be controlled through the applied magnetic field. In the present study, we demonstrate such a possibility through the study of an oxygen reduction reaction on a PdFe (001) surface by introducing a new concept: spin orientation dependent overpotential. Also, we have explained the origin of lower dissociation barrier for the O2 molecule on ferromagnetic surfaces when its spin moment is antiparallel to the surface magnetization as reported in the recent experiments.

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INTRODUCTION Oxygen-based electrochemical reactions, such as an oxygen reduction reaction (ORR) and oxygen evolution reaction (OER), which are extremely important for various applications including in fuel cells and batteries, are known to be limited due to their slow kinetics.1,2 The process of ORR involves the formation of nonmagnetic water molecules from paramagnetic oxygen in the gas phase, while OER produces paramagnetic oxygen molecules from nonmagnetic water.3,4 The chemical reactivity of oxygen to the catalyst surface is the most important factor in the overall process. In order to improve the sluggish reaction kinetics at the electrodes, efforts have been made in terms of manipulating various factors such as charge transfer and strain by introducing bimetallic alloys of Pt (platinum) and Pd (palladium), such as Pt(Pd)Fe, Pt(Pd)Co, Pt(Pd)Ni, etc., which are the best known catalysts for ORR.5−8 Because of their relatively lower costs as well as the ability to improve the catalytic activity through faster ORR activity, many researchers are focusing on studying the competence of these materials as replacements of Pt in proton exchange membrane fuel cells (PEMFC).9−12 The main reason for the growing interest in these alloy catalysts from a materials perspective is the charge transfer between the constituent metals.13 Because of the large electronegativity difference between the 3d (Fe, Co, Ni) and 5d (Pt,Pd) transition metals, electrons are transferred from the former to the latter, which is an important factor for tuning the chemical reactivity of oxygen to these surfaces.14,15 However, an overpotential as big as about 0.3 V is still observed,16 and efforts have been made to improve their activity. One aspect that has not been explored intensively so far is the role of magnetism. There are very few studies that have © XXXX American Chemical Society

connected the chemical bonding in heterogeneous interfaces of either the solid or gas phase17,18 to magnetic properties, even though, magnetic field induced enhancement of ORR were observed experimentally.19−23 The observed results are usually explained in terms of convection within the diffusion layer which again depends on the gradient of the magnetic field and the concentration gradient of the paramagnetic oxygen species.24 It should be noted that the results obtained in the present work do not require an applied magnetic field to have nonzero gradient and are valid in uniform magnetic fields. In the present study, we demonstrate that there is enough room for improving oxygen electro-catalysis through controlling the spin orientation of oxygen. In the present study, we focus on ORR. An analysis of the magnetic effects on OER in a similar manner can be extended easily.25 The reason for the sluggish ORR kinetics in the cathode of the PEMFC is the overbinding of the reaction intermediates, such as oxygen or hydroxyl, adsorbed on the cathode surface. For example, the typical dissociative adsorption has the following steps:1 1 O2 + * → O* 2 O* + (H+ + e−) → HO* HO* + (H+ + e−) → H 2O + *

(1)

Here, “∗” represents a catalyst site. It is known that the last two of the above equations are the rate limiting steps for the ORR.1 Received: October 13, 2017 Revised: December 13, 2017 Published: December 13, 2017 A

DOI: 10.1021/acs.jpcc.7b10147 J. Phys. Chem. C XXXX, XXX, XXX−XXX

Article

The Journal of Physical Chemistry C

different spin orientations. The adsorption energies were determined using the relation

The slow reaction kinetics is due to the overbinding of the reaction intermediates O* and HO* to the cathode surface. O* and HO* are very stable on the cathode surface and require additional potential (overpotential) to activate the last two steps, i.e., electron and proton transfer to form the water. A simple way to enhance the rates of these steps would be to destabilize slightly the surface bound O* and HO* in such a way that electron/proton transfer can occur easily. The main objective of this paper is to demonstrate that this can be done in a magnetic way as explained in the following section.

1 i y − jjjES ̂ + EO(g2) ↑ ( ↓ )zzz 2 k { 1 ij y g ( ) = ES +̂ O ↑ (↓) − jjES ̂ + EO2 zzz (2) 2 k { where “S” and “O’” represent, respectively, surface and oxygen. ̂ represents the surface spin while ↑(↓) represent the oxygen ̂

Eads, ↑ (↓) = ES

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+̂ O ↑ (↓)

̂

spin. Eads, ↑ (↓) is the spin dependent adsorption energy. E(g) O2 ↑ (↓) is the energy of the oxygen molecule in the gas phase and is independent of the direction of its spin. We used the PdFe (001) surface (with Pd termination) as a model for the ferromagnetic catalyst. The PdFe alloy has an L10 structure with alternating layers of Fe and Pd along the cdirection as shown in Figure 2. The surface was modeled using

METHOD As mentioned above, the first step is the dissociative adsorption of oxygen. If the catalyst on the cathode is nonmagnetic, the process of adsorption of oxygen cannot be controlled magnetically. However, if the catalyst is magnetic (such as PdFe or PtFe), the oxygen-surface interaction can be tuned through the magnetic field. Let us consider the following two possibilities. Case I. Suppose the oxygen prefers its spin to be antiparallel to the bimetal; in this case, if we apply a magnetic field in the direction of the easy axis of the bimetal, this would not affect the magnetization of the bimetal irrespective of the magnitude of the field. However, this will reorient the randomly oriented oxygen spins along the easy axis of the bimetal, resulting in a weaker binding of the bimetal. The situation is shown in Figure 1a. Indeed, this is the situation discussed in this paper.

Figure 2. PdFe (001) surface viewed along the c-axis: (a) with a larger O−O distance of (about 3.85 Å) and (b) with a smaller O−O distance (of about 2.76 Å).

periodic supercells containing slabs of their 2 × 2 in-plane unit cells and thickness of four atomic planes. All of the calculations were performed within the DFT framework with the Perdew− Burke−Ernzerhof functional26 of exchange correlation energy within a generalized gradient approximation and projector augmented wave (PAW) method as implemented in the Vienna ab initio simulation package (VASP).27 The top two atomic planes (surface layers) were relaxed, while the bottom two atomic planes were kept fixed in their bulk structure (bulk layers). The wave functions of valence electrons were expanded in a plane wave basis set truncated with an energy cutoff of 500 eV. The integrations over the Brillouin zone were sampled on a uniform grid of 5 × 5 × 1 k-points using Methfessel−Paxton smearing with a width of 0.1 eV. Ionic relaxation was performed such that the force on each ion was smaller than 0.03 eV/Å. The dipole correction was applied along the direction perpendicular to the metal surfaces to correct the electric fields arising from the structural asymmetry and periodicity. To obtain a desired spin orientation of the oxygen spin, we used the constrained DFT approach28,29 where the following energy functional was minimized:

Figure 1. Schematic representation of the relative orientation between the oxygen and slab magnetization for unconstrained (a) and constrained (b) cases. Arrows represent the directions of magnetization.

Case II. In another case, if the oxygen prefers its spin to be parallel to the bimetal, then one has to apply a magnetic field that is antiparallel to the easy axis of the bimetal. In this situation, the magnitude of the magnetic field must be smaller than the anisotropy field of the bimetal. Since the bimetal has large anisotropy, the magnetic field required to switch its magnetization is large, and therefore, a small magnetic field should be used that is just enough to reorient the oxygen spins without changing the magnetization of the bimetal. This situation is depicted in Figure 1b. To demonstrate how chemisorption depends on the magnetic arrangement, we performed first-principles calculation of the chemisorption energy of atomic oxygen for B


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orientation, we obtain smaller adsorption energies. The difference in the adsorption energy is larger for a small O−O distance (≃400 meV), while it is smaller for a slightly larger O−O distance (≃130 meV). The stronger distance (or coverage) dependence of the adsorption energy in this particular case arises due to the additional spin-dependent interaction among the oxygen atoms. The additional interaction could be the RKKY type32 interaction between two oxygen spins mediated via the conduction electrons of the metal surface.33 At a very small O−O distance, the oxygen spins are strongly exchange coupled. As the distance is slightly increased, this coupling weakens, and as a result, the strength of the coupling between the oxygen spins to the surface magnetization also changes. It is evident that by changing the spin orientation of the oxygen atom, one can decrease the strength of its binding to the surface. The next thing we would like to demonstrate is that it is easier to transfer electrons to this weakly bound oxygen. Let us consider the second term of eq 1 and write it in a slightly different manner

d

E[ρ , {M̂ I }] = EDFT [ρ] + Ep = EDFT [ρ] +

d

∑ λI (|MI| − M̂ I ·MI) (3)

I

The penalty energy term in eq 3 is given byEp = ∑IλI(|MI| − M̂ dI ·MI). M̂ dI is a unit vector along the desired direction of the magnetic moment. ρ is the electron density. The spin moment on the Ith atomic site, MI is defined by MI =

∫Ω m(r)FI(r) d r

(4)

I

FI(r) is a function of norm unity inside the sphere defined by ΩI and smoothly approaches to zero close to the boundary. λ is the weight of the penalty term. We used a value of λ in the range of 5−10 eV, which is quite reasonable for Fe-based systems. For all constrained magnetic calculations, we performed the relaxation of a few selected atoms (oxygen and top two layers of the slab) in order to ensure that the slab +oxygen system is close to the ground state.

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O*( ,̂ σ ) + (H+ + e−) → HO*( ,̂ σ ′)

RESULTS AND DISCUSSION In this study, we explore how the chemisorption of atomic oxygen on ferromagnetic surfaces is influenced by its relative spin orientation with respect to the metal surface. We have studied such spin orientation dependence of the chemisorption for different O−O distances, viz, the case when the O−O distance is about 2.76 Å and for an O−O distance of 3.85 Å. Oxygen atoms are placed on the hollow sites, which are usually considered to be the preferred site of adsorption on PdM (M = Fe, Co, Ni) surfaces. On the PdFe (001) surface, we note that Pd attains a magnetic moment of about 0.4 μB while Fe atoms have magnetic moments as large as 2.98 μB. All of the magnetic interactions (Fe−Fe, Fe−Pd, Pd−Pd) are ferromagnetic. It should noted that previous theoretical studies30,31 on Pt/ M(001) (M = Fe,Co) systems predicted substantially large transition temperature (above room temperature) for the ferromagnetic Pt layer. In the absence of any constraints, oxygen spins are antiparallel to the surface magnetizations. To find the effect of spin orientation, the oxygen spins are aligned parallel to the magnetization of the metal atoms through the penalty term as indicated in eq 3. In Table 1, we show the calculated adsorption energies for different relative orientations between the surface magnetization and oxygen spin at different coverages. By constraining the oxygen spins in a direction different from its equilibrium

where σ = and σ′ are the spin configurations of the surface bound oxygen and hydroxyl molecule, respectively. The change in the free energy is given by, ΔGσ , σ′ = G(HO*( ,̂ σ ′)) − G(O*( ,̂ σ )) − G(H + + e−)

coverage

2.76

0.5

3.85

0.25

surface−oxygen relative spin orientation

Eads, ↑ (↓) (eV)

̂

ZPEEO* (eV)

̂ ( ,↑) ̂ ( ,↓) ̂ ( ,↑)

−3.60

0.04

−4.01

0.08

−3.57

0.05

̂ ( ,↓)

−3.70

0.03

(6)

The more negative the change in free energy, as represented by eq 6, the more easily the electron can be transferred to the adsorbed oxygen. The results are shown for a small O−O distance (i.e., 2.76 Å). We also calculated the adsorption energy of the hydroxyl (OH) ion for this particular coverage. Again, in the absence of constraints, OH prefers to bind to the surface with its spin antiparallel to the surface magnetization. The adsorption energy of OH with its spin antiparallel to the surface magnetization is −2.23 eV while it is −2.03 eV for the constrained (parallel) case. The weaker dependence of the adsorption energy on the direction of spin in this case is due to the smaller magnitude of the spin moment of OH. Without any applied potential, the change in the free energies for successive steps can be written as ΔG = ΔE + ΔZPE − T ΔS

(7)

Here, ΔE is the change in the reaction energies obtained from the DFT calculation. ΔZPE and TΔS are the changes in the zero point energies and entropies in different reaction steps. The zero point energies ZPEs for H2, O2, O*, and HO* are calculated from vibrational frequencies obtained using the finite difference method. For O* and HO*, we only considered the adsorbate modes (the slab atoms were considered to be frozen). The displacements along each direction with steps of 0.015 Å were considered. The calculated ZPEs of adsorbed oxygen for different spin configurations and coverages are shown in Table 1. The entropic data are taken from the standard tables for gas phase molecules.34 We build the free energy diagram for the surface with small 1 O−O distance(2.76 Å). We use the following states: H2 + 2 O2 ,

Table 1. Calculated Spin Orientation Dependent ̂ and Anti-Parallel Adsorption Energies for Parallel ( ,↑) ̂ ( ,↓) Surface−Molecule Spin Configurationsa O−O distance (Å)

(5)

1

H2 + O*, 2 H2 + HO* and H2O, which are shown in Figure 3. The reference potential is set to the standard hydrogen potential as proposed by Nørskov et al.,1 according to which

a ̂ represents surface spin while ↑/↓ represents oxygen spin. The zero point vibrational energy of adsorbed oxygen is also shown.

C


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Figure 3. Potential dependent free energy diagram for ORR. The results are shown at zero potential (U = 0 V, solid lines), onset potential (U = 1.0 and 0.8 V, dot dashed lines), and at equilibrium potential (U = 1.23 V, dashed line). Blue (red) lines correspond with (without) constrains on magnetic moments.

Figure 4. Charge density differences Δρ calculation) from eq 10 are shown.

,̂ ↑

(z) (blue line, for constrained magnetic calculation) and Δρ

for an electrode potential U = 0 at T = 298 K, pH = 0, and equilibrium between the electrolyte and 1 bar of H2, the free 1 energy of (H+ + e−) is equal to the free energy of 2 H2 . It should be noted that the effect of the charging of the electrode or presence of counterions were not included35 in our study. Without any applied potential, all of the steps are exothermic for both the unconstrained and constrained cases. The formation of reaction intermediates are more downhill for the unconstrained case with respect to the constrained one showing that in the absence of any external potential, the antiparallel arrangement between the magnetization of the reaction intermediates and the surface magnetization is thermodynamically preferable. In order to determine the effect of the cell potential on the reaction steps, we shift all the steps involving electron and proton transfer by an amount -eU,

,̂ ↓

(z) (red line, for unconstrained

where e is the electronic charge and U is the applied potential. Using such a scheme, we obtain spin-orientation dependent overpotantials defined as follows, η( ,̂ ↓ ) = U0 − U ( ,̂ ↓ ) η( ,̂ ↑ ) = U0 − U ( ,̂ ↑ )

(8)

̂ is the overpotential for the unconstrained where η( ,↓) ̂ is the corresponding one for the (antiparallel) case and η( ,↑) constrained case. U0 is the equilibrium potential having a ̂ standard value of 1.23 V for the fuel cells. U( ,σ(=↑,↓)) are the onset potentials or the maximum potentials at which the each step of the reaction is downhill. The onset potential for ̂ and unconstrained ( ,↓) ̂ cases are the constrained (U( ,↑) D


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The Journal of Physical Chemistry C respectively 1.0 and 0.8 V as shown in the Figure 3. ̂ = 0.43 Corresponding overpotentials are respectively η( ,↓) ̂ eV and η( ,↑) = 0.2 eV. The overpotential is therefore reduced by about 46% (0.23 eV) when the constraint is applied by fixing the spin orientation of oxygen. When we shift the chemical potential of the electrons by a bias of 1.23 eV, it can be seen that, the last of the two steps shown in eq 1 are then uphill. The rate-determining step being the reduction of O* to OH* for any relative spin orientation between the reaction intermediates and the surface. Here we point out that, as the effect of the solvent was not considered, the reduction in overpotential might have been slightly overestimated. To understand the nature of bonding for different spin arrangements between the surface and oxygen, we consider two charge density differences given by Δρ

,̂ ↑

(x , y , z) = ρslab + O ( ,̂ ↑ ) − ρslab − ρO

Δρ

,̂ ↓

(x , y , z) = ρslab + O ( ,̂ ↓ ) − ρslab − ρO

Figure 5. LDOS of the surface Pd-atoms and O are shown for the constrained and unconstrained cases. (9)

for parallel and antiparallel magnetic configurations between the surface and the adsorbed oxygen, respectively, where ρslab ̂ are respectively the charge densities of the and ρslab+O( ,σ) bare FePd slab and the slab with oxygen in a spin orientation σ adsorbed in it. ρO is charge density of the oxygen atom in the gas phase. After performing a planar integration, we obtain the charge densities along the z-direction. =

∫

(z ) =

∫

Δρ

(z ) ,̂ ↑

Δρ

,̂ ↓

z

d z′

∫ ∫ dx dy Δρ

,̂ ↑

(x , y , z′)

d z′

∫ ∫ dx dy Δρ

,̂ ↓

(x , y , z′)

z

smaller than the field required to switch the magnetic orientation of the catalyst. In the present study we have demonstrated the effect of spin orientation on ORR in the context of dissociative mechanism. In reality ORR can also proceed through an associative path way, for example, which involve peroxyl (OOH*) dissociation instead of O2 dissociation. However, depending on the potential, the rate-determining step could be the same as that of dissociative path (i,e the reduction of O* to OH*).36 Also it was pointed out that,1 at realistic potentials both dissociative and associative mechanisms can run in parallel. Recently Duan et al.37 have compared three mechanisms of ORR on the Pt(111) surface: oxygen dissociation, peroxyl dissociation, and hydrogen peroxide dissociation. They concluded that ORR would follow the peroxyl dissociation path with a rate-determining step involving reduction of O* to OH* as it is in the case of the present study.

(10)

The obtained results are shown in Figure 4, where values of Δρ ,̂ ↑ (z)(z) and Δρ ,̂ ↓ (z) are plotted for the high oxygen coverage case. In the unconstrained case, more electrons are accumulated between the surface and the oxygen atom (red curve, peak around 12.6 Å), while in the constrained case, as expected, fewer electrons are accumulated (blue curve, peak around 12.2 Å). The slight difference in the locations of peaks is due to the different relaxations in the two cases (constrained, unconstrained). The difference in bonding between oxygen and the surface can be further understood in terms of local density of states (LDOS), as shown in Figure 5, where the LDOS of Pd atoms of the surface layers projected on d-states is shown alongside with the LDOS of the oxygen atom projected on p-states. For the unconstrained case (lower panel), there is a strong bonding peak around −5.5 eV, which is absent in the constrained case, resulting in a weaker binding. Our calculation therefore suggests that it is possible to control the chemical kinetics of oxygen-based electrochemical reactions such as ORR and OER through the magnetic field. By orientating the magnetization of the reaction intermediates through the penalty energy term shown in eq 3, an additional

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EFFECT OF SPIN ORIENTATION ON THE DISSOCIATION OF THE O*2 MOLECULE In the free energy diagram, we have not included the surface bound O2* molecule, as we have treated it to be a reference state in the solution. However, we have studied the effect of spin orientation on the dissociation of the surface bound O*2 molecule as our results discussed in terms of O2 dissociation. The experimental work of Kurahashi et al. suggests that on a ferromagnetic surface such as Ni(111), the O2 molecule shows a higher sticking probability when its spin moment is antiparallel to the surface moments.38 Also, they observed that such antiparallel spin arrangement of O2* molecule with respect to the surface magnetization leads higher dissociation rate. To understand the effect of the spin orientation on the O2* dissociation, we calculated the transition state (TS) energies using the fixed bond length method39 within the constrained DFT formalism. The activation barriers Ea are defined through, Ea = ETS − EIS, where ETS and EIS are respectively energy of the transition and initial state. We have considered the following two reactions:

∂Ep

magnetic field Bcon = − ∂m(r) is introduced in the simulation,

which mimics the effect of the magnetic field for the real case. The primary requirements are that the catalysts should be (1) ferromagnetic and (2) magnetically hard, so that the applied magnetic field should be sufficient enough to rotate the spin directions of the O-species (O and OH) so that they are less stable on the cathode surface and the barrier for electron and proton transfer is smaller. However, the applied field should be

O*2 ( ,̂ ↑ ↑ ) → O*( ,̂ ↑ ) + O*( ,̂ ↑ ) O*2 ( ,̂ ↓ ↓ ) → O*( ,̂ ↓ ) + O*( ,̂ ↓ ) E

(11)


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The Journal of Physical Chemistry C The first of the eq 11 refers to dissociation of a surface bound O2* molecule in the spin triplet state with its spin parallel to the surface magnetization while the second part of eq 11 refers to when its spin is antiparallel to the surface magnetization. The choice of the initial spin states in eq 11 are in accordance with the recent experiments by Kurahashi et al.,38 which suggest that the O2 molecule retains its spin triplet state very close to the ferromagnetic surface (close to 2−3 Å). The calculated activation energies are 1.16 and 1.0 eV, respectively, for the spin parallel and antiparallel case respectively as can be seen from the Table 2. To understand the origin of the lower

be transferred. Although, for a ferromagnetic surface states near the vacuum are dominated by the minority electrons, yet there are finite majority carriers present (unless the surface is an half-metal). It is interesting to notice that the dissociation of O2 pass through a TS which is allows transfer of both type of carriers only when its spin is antiparallel to the surface. This explains the origin of the lower dissociation barrier for the antiparallel spin arrangement between the surface and molecule as observed in recent experiments.38

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CONCLUSION In conclusion, we showed that oxygen-based electrochemical reactions such as ORR can be controlled through magnetism. The binding of reaction intermediates to the surface can be made weaker by constraining their spin direction; hence, these relatively unstable reaction intermediates will be more prone to electron and proton transfer processes. In a real experiment, this might be done using a weak magnetic field that can polarize the spin orientations of the reaction intermediates in a manner unfavorable for strong chemisorption but that does not affect the magnetization of the ferromagnetic catalyst nanoparticles. We have also analyzed the transition states (TS) for the dissociation of the surface bound O2 molecule and the reason behind its higher dissociation rate when its spin is antiparallel to the surface magnetization.

Table 2. Energy Barrier (Ea), Reaction Energy (ΔH) and Charge Transferred from the Surface to the Molecule at the TS for Different Reactions reaction

Ea (eV)

ΔH (eV)

δQO2*# (e)

̂ ̂ + O*( ,↑) ̂ O*2 ( ,↑↑) → O*( ,↑) ̂ ̂ ̂ O2*( ,↓↓) → O*( ,↓) + O*( ,↓)

1.5

0.26

+1.0

1.0

0.15

+1.6

dissociation barrier for the antiparallel spin configuration, we have studied the charge transfer from the surface to O2* using Bader charge analysis.40 In the Table 2, we also show the change in Bader charge of the adsorbed molecule at the transition state (given by δQO2*#). The Bader charge at the TS show that the second of the eq 11 is more favorable as the charge transfer from the surface to O*2 is more efficient. To understand the exact role played by the charge transfer effect in such dissociative adsorption, we compared the density of states of projected on p-states (pDOS) for three different cases:O2 molecule at (a) free state, (b) at the TS for the process represented by the first of the eq 11 (spin-parallel case) and (c) at the TS for the process represented by the second of the eq 11 (spin-antiparallel case). The results are shown in the Figure 6. The transition state for the antiparallel case has very interesting features in comparison to the other two cases. In this case lowest unoccupied molecular orbital (LUMO) allows both spin majority and minority electrons from the surface to

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AUTHOR INFORMATION

Corresponding Author

*(S.-C.L.) E-mail: [email protected]. ORCID

Seung-Cheol Lee: 0000-0002-9741-6955 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by NRF grants funded by MSIP, Korea (No. 2009-0082471 and No. 2014R1A2A2A04003865), the Convergence Agenda Program (CAP) of the Korea

Figure 6. Density of states of the O2 molecule projected on the p-states for three different cases: (a)free, at the TS state for the (b) first and (c) second of the reactions described by eq 11. It can be seen that the LUMO in part c contains both majority and minority spin states. F


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Research Council of Fundamental Science and Technology (KRCF), and the GKP (Global Knowledge Platform) project of the Ministry of Science, ICT, and Future Planning.

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Article

The Journal of Physical Chemistry C (40) Tang, W.; Sanville, E.; Henkelman, G. A Grid-based Bader Analysis Algorithm Without Lattice Bias. J. Phys.: Condens. Matter 2009, 21, 084204.

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