Abstract
The dband center model of Hammer and Nørskov is widely used in understanding and predicting catalytic activity on transition metal (TM) surfaces. Here, we demonstrate that this model is inadequate for capturing the complete catalytic activity of the magnetically polarized TM surfaces and propose its generalization. We validate the generalized model through comparison of adsorption energies of the NH_{3} molecule on the surfaces of 3d TMs (V, Cr, Mn, Fe, Co, Ni, Cu and Zn) determined with spinpolarized density functional theory (DFT)based methods with the predictions of our model. Compared to the conventional dband model, where the nature of the metaladsorbate interaction is entirely determined through the energy and the occupation of the dband center, we emphasize that for the surfaces with high spin polarization, the metaladsorbate system can be stabilized through a competition of the spindependent metaladsorbate interactions.
Introduction
Due to the low abundance, toxicityrelated issues and high cost of 4d and 5d metals, in recent years researchers have turned to developing catalysts using cheap and abundant 3d transition metals (TMs) and their alloys or oxides^{1,2,3}. The catalytic reactions in these materials can also be manipulated using spin, in addition to the usual parameters such as size, strain, and electrode potential. The role of magnetism in heterogeneous catalysis is the subject of a recent study^{4,5,6,7,8}. It was demonstrated by Behler et al.^{9} that on metal surfaces such as Al (111), spin selection leads to a low sticking probability of O_{2} molecules with a triplet spin state. Recently, Melander et al.^{10} showed that the reactivity of metal surfaces is dependent on their magnetic states. Using firstprinciples methods, they noted that in the case of adsorption of H_{2} on a ferromagnetic Fe surface, there is an asymmetry in the FeH_{2} interaction for majority and minority spin channels. Such asymmetric interaction results in weaker hydrogenmetal binding for a ferromagnetic Fe surface than for an antiferromagnetic Fe surface. In the ferromagnetic case, only spin minority electrons take part in the bond formation, while on the antiferromagnetic surface, the bond formation is accomplished through both the minority and majority spin electrons.
Such notable results obtained either from the firstprinciples simulations or experiments require a simple theoretical model to interpret. The majority of the firstprinciples theoretical studies are focused on the understanding the nature of interaction between the adsorbate and the delectrons of the TM surface^{11,12,13,14,15,16}. The most widely employed model invoked to understand the role of the delectrons is the socalled dband center model^{17,18,19,20,21} of Hammer and Nørskov, developed more than a decade ago. This simple yet highly celebrated model of chemisorption is again based on the concepts of other models of chemisorption such as (1) the NewnsAnderson model^{22,23} and (2) the effective medium theory^{24,25,26}. The former is a more general description of the interaction of the adsorbate state with the continuous band of valence states of the metal, while the latter relates the adsorption energy to the local electron density and the change in oneelectron states of the surface.
In the dband model, the band of dstates participating in the interaction is approximated with a single state at energy ε_{d}, known as the center of the dband. Such a model can be thought of as a narrow dband limit of the NewnsAnderson model. According to this model, the variation in the adsorption energy from one TM surface to another correlates the upward shift of this dband center with respect to the Fermi energy. A stronger upward shift indicates the possibility of the formation of a larger number of empty antibonding states, leading to a stronger binding energy. The upward shift of the dband center can therefore be treated as a descriptor of the catalysis. HammerNørskov model successfully explains both the experimental and the firstprinciples theoretical results for different ligands/molecules on a variety of TM surfaces^{27,28,29}.
However, there are few studies on the adsorption of molecules on metal surfaces with high spin polarization. Moreover, if an adsorbate itself has a considerable magnetic dipole moment, it will have a strong magnetic interaction with the surface. Therefore, the validity of the HammerNørskov model for molecular adsorption on surfaces with large spin polarization is not obvious. The dband center model predicts a uniform decrease (increase) of the adsorption energy of a given molecule from one TM surface to another where the number of delectrons increases (decreases). An exception to the prediction of the dband center model occurs for OH adsorption on Pt and Pd skin alloy systems^{30}. However, such exceptions are typically related to the large electronegativity of the adsorbate and the substrate having a nearly full dband.
In the present study, we demonstrate the limitations of the conventional dband center model via a simple case study: the adsorption of nonmagnetic molecules such as NH_{3} on 3d TM surfaces. The reaction of NH_{3} on TM surfaces is important due to its relevance in controlling the corrosion of steel and iron surfaces. We show that for a better comparison with the results obtained from the spinpolarized DFTbased methods, the conventional dband center model has to be extended by considering two band centers, one each for the spin majority and the spin minority electrons of the system. Such a model would be useful in designing chemical reactions that can be controlled through spin arrangement of the catalytic surface or by an external magnetic field.
Adsorption on 3dTM surfaces; why do we need a spinpolarized dband center model?
Here, we examine the applicability of the conventional dband center model to a simpler problem: adsorption of nonmagnetic molecules on spinpolarized metal surfaces. From a comparison of the adsorption energies of an NH_{3} molecule on 3d TMs obtained from spinpolarized and spinunpolarized calculations, we find a significant effect of spin polarization on adsorption. The adsorption energies of the molecule on magnetic surfaces are smaller for the spinpolarized calculations (see Fig. 1). This simple fact also suggests that the dband center model, which relies on a nonspinpolarized (or spinaveraged) description of the surface electrons, has to be expanded to incorporate spin polarization effects.
To understand the trend in catalytic activity across TMs, one should consider the spin polarization of the metal surface in addition to the number of delectrons. In Fig. 2, we schematically compare the dband center of a metallic surface with and without spin polarization. When spin polarization is considered in a calculation, it is appropriate to consider two dband centers, one for the spin up states ε_{d↑} and the other for the spin down states ε_{d↓}. These are shifted in opposite directions in energy relative to the unpolarized dband center, ε_{d}. ε_{d↑} is shifted downwards, while the ε_{d↓} is shifted upwards with respect to ε_{d}. If we consider that these two centers interact with the adsorbate level, we should obtain two sets of bonding and antibonding orbitals that are higher and lower in energy with respect to the unpolarized bonding and antibonding levels. The possibility of obtaining a nonlinear dependence of the adsorption energy with the number of delectrons originates from the fact that the contributions to the adsorption energy from two such band centers can compete with each other. Naturally, when the degree of the spin polarization is smaller, the two dband centers are close to each other, and their activity is similar. However, when the spin polarization is higher, the two band centers are shifted significantly in opposite directions. If we consider the interaction with an adsorbate possessing multiple levels, among which the occupied ones are closer to the metal band centers than for the minority spin, there are more unoccupied metaladsorbate antibonding states giving rise to strong attractive interactions, while there are more occupied metaladsorbate states for the majority spin electrons, resulting in strong repulsion. Therefore, the minority spin dbands bind more strongly to the adsorbate, while the binding with majority spin states is weaker. This phenomenon results in large changes in the adsorption energies of Mn and Fe as shown in Fig. 1 for spinpolarized and nonspinpolarized cases.
Twocentered dband model
In this section, we generalize the dband model but still follow the approach used by Hammer and Nørskov. Let us consider the interaction of the adsorbate states with the metal states using a basis set with a minimum number of states, {ψ_{aiσ}, ψ_{dσ}}, where ψ_{aiσ} is the i^{th} adsorbate state with spin σ and ψ_{dσ}(σ = ↑, ↓) are two hypothetical discrete states representing metal states with two spins. The adsorption energy can be expressed as follows (see the supplemental material):
For simplicity, we have assumed that all the adsorbate states are sigmatype orbitals. are the matrix elements of the coupling between the TM dstate with the k^{th} adsorbate state, is the energy of the i^{th} unoccupied adsorbate state with spin σ′ and ε_{ajσ′} is the energy of the j^{th} occupied adsorbate state. The two dband centers for the majority and minority spins are respectively ε_{d↑} and ε_{d↓}. The first term in the above equation is the energy gain due to the interaction of the unfilled adsorbate state with the metal states. The second term describes the interaction of the metal dstates with the filled adsorbate states. The first term always describes an attractive interaction, while the second term has both attractive and repulsive components. Here, f_{σ} is the fractional filling of the metal state with spin σ. The last two terms in Eqn. (1) are due to the orthogonalization of the adsorbate state and TM dstates and are always repulsive. The parameter α is adjustable and has units of eV^{−1}. The third term is due to the orthogonalization of the empty adsorbate states on the metal dstates, while the fourth term represents the orthogonalization of the filled adsorbate states on the metal states.
When there is more than one adsorbate state, with some filled and some empty, to understand how the net attractive and repulsive interactions compete with each other in a realistic situation, we consider the case of an NH_{3} molecule on the TM surface. In this case, the adsorbate is nonmagnetic, and we assume that the interaction parameter is spinindependent and constant for a particular metal surface. We express the various energy contributions to the molecule and delectron interaction as follows:
where N and M are respectively the number of unoccupied and occupied adsorbate orbitals.
Spindependent attractive and repulsive surfaceadsorbate interaction
Eqn. (2) describes a simplified model for adsorption energy of a nonmagnetic molecule interacting with the TM surface. The states with energy and ε_{aj} are respectively empty antibonding and filled bonding molecular states. Competition and cooperation between the different spin channels during the process of adsorption are evident as we split the energy given by Eqn. (2) into attractive and repulsive parts. The first term in Eqn. (2) is always attractive for arbitrary filling of the dstates for both the spins, while the second term can be written as the sum of attractive and repulsive contributions. The attractive component is as follows:
The first term of Eqn. (3) gives the gain in energy due to the empty adsorbate levels interacting with the dband centers, while the second term is the energy gain due to the bonding orbitals formed between the filled adsorbate states and the dband centers. The energy due to the repulsive interaction between the molecule and the metal surface is given as follows:
The first term of Eqn. (4) is the energy of the antibonding orbitals, which promotes destabilization of the adsorbate on the metal surface, while the last two terms result from the orthogonalization of the metal and adsorbate states, as already mentioned.
Results and Discussions
In this section, we quantify the energy contributions mentioned in Eqns (2, 3 and 4). We have calculated the matrix elements V for different TM surfaces using Harrisons rule^{31}:
where , are constants. The characteristic length r_{d} of the dorbitals of different TM atoms is taken from ref. 31. The bondlengths d were taken from our DFT calculations. Because no πtype molecular orbital is involved, we have considered only (note that here, σ indicates the type of adsorbate orbital, not the spin index, as in Eqn. (2)). To calculate the attractive and repulsive contributions in Eqns (3 and 4) of an NH_{3} molecule on different TM surfaces, we considered four discrete energy levels of the NH_{3} molecule (obtained from the DFT calculations) in a symmetric manner, two from the HOMO region and two from the LUMO region, (see Fig. 3, where the density of states (DOS) of the NH_{3} molecule is shown). The DOS exhibits five distinct peaks at the energies = −15.4 eV, −5.5 eV, and −0.5 eV and = 4.4 eV, and 6.4 eV, respectively. Among these peaks, the peak at −5.5 eV corresponds to the doubly degenerate NH bonding molecular orbital with 1e symmetry, while the peak at −0.5 is due to the molecular orbital with 3a1 symmetry representing the lone pair. The peaks at 4.4 eV and 6.4 eV are the antibonding molecular states with symmetries 4a1 and 2e, respectively. In our calculation of the chemisorption energy, we have not considered the level at −15.4 eV since it is energetically too far from both the majority spin and the minority spin dband centers for all the TMs.
The adsorption energies are calculated from Eqn. (2), where the renormalized adsorbate levels ε_{aj} and are due to the interaction with sp electrons of the metal. These levels are obtained using the NewnsAnderson model^{22,23} (see Fig. 4, where the renormalized levels are shown alongside the dprojected DOS of the Fe (110) surface). The corresponding renormalized DOS of the NH_{3} molecule is as follows:
where is the chemisorption function. describes the adsorbatemetal coupling for the sp electrons, and D_{sp}(E) is the DOS of the metals sp electrons. is the KramersKrönig transformation of Δ(E). The renormalized adsorbate levels are calculated from the values of E for which the lines described by cross Λ(E).
In the actual calculation of Δ(E), we assume a semielliptical sp band centered at the Fermi energy, with the bandwidth obtained from our DFT calculation.
In Table 1, in the 2nd and 3rd columns, we show calculated dband centers for the majority spin and the minority spin for TM surfaces in the 3d series. The fourth column gives the attractive contribution to the metalligand interaction, while the fifth column gives the repulsive part of the metalligand interaction. Table 1 shows that for V, Cr, Cu and Zn, which is not the case for Mn, Fe, Co, and Ni, for which ε_{d↑} < ε_{d↓}. The 6th and 7th columns give the magnitude of the spindependent attractive interaction, while the 8th and 9th columns give the magnitude of the spindependent repulsive interaction. It is evident from the table that for V, Cr, Cu and Zn, the energies for the attractive interaction are the same for both the majority and minority spins. Additionally, as expected, the energies for the repulsive interaction are the same for both the spins. In contrast, for Mn, Fe, Co, and Ni (see columns 8 and 9 of Table 1), the attractive interaction has a larger magnitude for the minority spin, while the repulsive interaction has a larger energy for the majority spin.
In the case of NH_{3}, the strongest moleculeTM interaction is through the filled lone pair^{15,32}. For spinpolarized surfaces, most of the repulsive interaction is produced by the majority spin electrons, mainly because (1 + f_{↑})αV^{2} > (1 + f_{↓})αV^{2} since f_{↑} > f_{↓}.
In Fig. 5, we show the adsorption energies obtained from the spinpolarized DFT calculations alongside the ΔE_{d} calculated from our model (left panel) using Eqns (3) and (4). For comparison, we also show ΔE_{d} calculated from the HammerNørskov model (right panel). The dband centers in this case were obtained from the spinunpolarized DFT calculations. It is evident from Fig. 5 that our model is more consistent with the trend of the adsorption energies representing the DFT calculation. This better fit arises because the spindependent competing metaladsorbate interaction (which is important for Mn, Fe, Co, and Ni) is absent in the HammerNørskov model.
Instead of the spinaveraged dband center, , we propose that the adsorption energies obtained from the spinpolarized DFT calculation (or from the experiments) can be correlated with the following descriptor:
where is the reduced fractional moment. The first term is the usual spinaveraged dband center, while the second term is a shift depending on the spin polarization of the surface. The second term is nonzero only for the surfaces with a nonzero magnetic moment. The role of this term is to push the effective dband center to lower energy and thus capture the effect of the spin polarization in reducing the adsorption energy. For f_{↑} = f_{↓} and ε_{d↑} = ε_{d↓} = ε_{d}, ε^{eff} = ε_{d}, the descriptor for the usual dband center model. In Fig. 6, we plot ε^{eff} with the adsorption energies obtained through spinpolarized DFT calculations and show the spinaveraged dband center for comparison.
General relationship between chemisorption energy and dband centers in spinpolarized systems
The variation of the chemisorption energy from one metal surface to another as predicted in the conventional dband center model^{33} is as follows:
The first term in Eqn. (8) corresponds to the covalent interaction between the metal and the adsorbate, while the second term corresponds to the Pauli repulsion due to orthogonalization^{34} of TM and adsorbate states. , while . ε_{d} is the dband center that is obtained either from a nonspinpolarized calculation or through spin averaging, . Ignoring the second term, we obtain the following:
Eqn. (9) represents the central result of the conventional dband center model^{17,34}, i.e., a positive shift in δε_{d} implies an increase in the chemisorption energy, while a negative shift in δε_{d} decreases the chemisorption energy.
The variation of the chemisorption energy and the dband center has the following relationship from our spingeneralized model from Eqn. (2):
where and .
The form of Eqn. (10) suggests a decrease in the chemisorption energy as we move from a minimally spinpolarized surface to a highly spinpolarized one, since if we consider δε_{d↑} to be positive, δε_{d↓} should be negative, and the first two terms in Eqn. (10) will compete. The change of the chemical reactivity due to the antiferromagnetictoferromagnetic crossover^{10} can also be understood in terms of Eqn. (10). For antiferromagnets, there are two spin sublattices, which we label A and B. If we consider the simplest case, in which both of the sublattices are composed of the same metal, we have
From Eqn. 11, the stronger coupling of an adsorbate to the minority spin channel of the sublattice A implies a strong coupling to the majority spin channel of the sublattice B. This coupling can lead a change of site preference, even in a monocomponent antiferromagnetic material.
Stoner criterion and chemisorption
The formation of local moment on the i^{th} site of a TM surface is governed by the local Stoner criterion, D_{i}(E_{F})I > 1, where D_{i}(E_{F}) is the DOS of the delectrons on i^{th} site at the Fermi energy and I is the Stoner integral. Since strong chemisorption pushes a large number of states from the region near the Fermi energy to lower energies (due to bond formation with the adsorbate), it therefore disturbs the Stoner criterion locally. Thus, these two effects, viz, chemisorption and the Stoner criterion, oppose each other. The former leads to an increase in the kinetic energy, while the latter promotes a smaller kinetic energy so that the magnetism is retained. It is therefore expected that the spinpolarized surfaces would show lower activity than the nonspinpolarized surfaces.
Outlook
It should be noted here that this approach of considering multiple dband centers can be further extended to study catalytic reactions involving TM oxides, which will help us design inexpensive catalysts^{3}. The dbands of such systems are usually not continuous and contain multiple subbands, mainly due to the crystal field effect. The number and the arrangement of such subbands depends on the symmetry of the crystal field. If the system is magnetic, these subbands further split into minority and majority spin subbands. A reliable description of the catalytic activity of such systems can be obtained only from a model with a Hamiltonian of , which describes the interaction between the adsorbate levels ε_{aσ} with a set {i} of spindependent dband centers {ε_{diσ}} with occupations n_{diσ}. are respectively the creation and annihilation operators for the adsorbate states, while are the corresponding operators for the dstates. For perovskites with an ABO_{3} structure, i ∈ t_{2g}, e_{g}. Additionally, using the present approach allows one to investigate how to activate the reactions that are forbidden due to conservation of the spin angular momentum^{5}, by choosing a catalyst material with appropriate spin polarization. Although socalled twostate reactivity has already been the subject of a case study of organometallic complex catalysts^{35}, the concept was not discussed rigorously for heterogeneous catalysts, most importantly using the concept of dband centers (narrow dband limit). There are catalytic reactions in which both the reactants and the products are nonmagnetic, but the reaction intermediates can be magnetic, and the ratedetermining steps can depend on the spin exchange between the adsorbate and the surface. A more complete analysis along this direction is a subject of future studies.
Methods
The adsorption energies and the spindependent band centers are calculated from first principles. These firstprinciples calculations are performed within the framework of DFT with the PerdewBurke Ernzerhof exchange correlation energy functional^{36} based on a generalized gradient approximation. We used a projector augmented wave method as implemented in Vienna ab initio simulation package (VASP)^{37}. The surfaces were modelled as slabs of 4 × 4 inplane unit cells and four atomic layers containing 64 atoms. KohnSham wave functions of the valence electrons were expanded in a plane wave basis with an energy cutoff value of 450 eV. Brillouin zone sampling was conducted using a Monkhorst Pack grid of 3 × 3 × 1 kpoints. Ionic relaxation was performed using the conjugategradient method until forces were reduced to within 0.02 eV/Angstrom for the nonconstrained atoms. A vacuum of 10 Å was included. In all cases, we considered closepacked structures of TM surfaces. We considered ferromagnetic (011) surfaces of V, Cr, Mn and Fe, the (0001) surface of Co, and the (111) surfaces of fcc Ni, Cu and Zn. The dipole corrections were applied along the directions perpendicular to the metal surface to eliminate the unwanted electric fields arising from the asymmetry of the simulation cell. The structural relaxations were performed for NH_{3} and only the top two layers of the TM surface. The bottom two layers were fixed to their bulk experimental values. The adsorption energy was calculated from the following relation:
where E_{S+A} is the energy of the surface plus the adsorbate and E_{S} and E_{A} are the energy of the surface and adsorbate, respectively. We used Eqn. (12) to calculate the adsorption energies of NH_{3} on different TM surfaces with and without spin polarization. The dband centers of both the majority spins and the minority spins were calculated from the first moment as given by^{19},
where D_{dσ}(E) is the DOS projected on the dstates of the TM for spin σ and E_{F} is the Fermi energy of the system. The spindependent fractional occupations are considered as follows: . These band centers and occupations were used as inputs for Eqns (2, 3 and 4)^{38}.
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How to cite this article: Bhattacharjee, S. et al. An improved dband model of the catalytic activity of magnetic transition metal surfaces. Sci. Rep. 6, 35916; doi: 10.1038/srep35916 (2016).
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Acknowledgements
UVW acknowledges funding from the IndoKorea Institute of Science and Technology and support from a J.C. Bose National Fellowship (Dept. of Science and Technology, Govt. of India). This work was supported by the Convergence Agenda Program (CAP) of the Korea Research Council of Fundamental Science and Technology (KRCF).
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S.B. and S.C.L. conceived the idea. S.B. performed the numerical and analytical calculations, wrote the the paper. S.C.L. and U.V.W. provided valuable input.
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Bhattacharjee, S., Waghmare, U. & Lee, SC. An improved dband model of the catalytic activity of magnetic transition metal surfaces. Sci Rep 6, 35916 (2016). https://doi.org/10.1038/srep35916
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