Structural, Magnetic, Electronic, Defect, and Diffusion Properties of

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The Structural, Magnetic, Electronic, Defect and Diffusion Properties of CrO : a DFT+U Study 2

3

François Lebreau, Mazharul M Islam, Boubakar Diawara, and Philippe Marcus J. Phys. Chem. C, Just Accepted Manuscript • Publication Date (Web): 20 Jun 2014 Downloaded from http://pubs.acs.org on June 23, 2014

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The Journal of Physical Chemistry

The Structural, Magnetic, Electronic, Defect and Diffusion Properties of Cr2O3: A DFT+U Study François Lebreau, Mazharul M. Islam,, Boubakar Diawara*, Philippe Marcus Institut de Recherche de Chimie Paris, CNRS – Chimie Paristech, 11 rue Pierre et Marie Curie, 75005 Paris, France.

Abstract

The structural, electronic and magnetic properties for non-defective and defective structures and the diffusion of both cations and anions in chromium oxide (α-Cr2O3) are investigated theoretically with periodic quantum-chemical method. Three different point defects are studied namely Cr vacancy, Cr Frenkel defect (composed of an interstitial Cr atom and a Cr vacancy) and O vacancy. All these defects affect the electronic properties of Cr2O3 drastically and are involved in diffusion processes in passive film growth. The calculated defect formation energy shows that the stability of defects falls in the following order: Cr Frenkel defect (EFr(Cr) = 2.36 eV) > Cr vacancy (EV(Cr) = 4.84 eV) > O vacancy (EV(O) = 5.12 eV). Relaxation occurs only on the first and the second nearest-neighbors in each case. Each defect adds an extra localized level inside the band gap. Cr Frenkel defects add donor levels composed of O states, Cr vacancy defects add acceptor levels composed of states from both Cr and O atoms, and O vacancies do

*

Corresponding author: [email protected] 1 ACS Paragon Plus Environment


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not give any level in the gap. Defects influence the magnetic moments on surrounding atoms, especially on the first nearest-neighbors. Various diffusion processes of both cations and anions are investigated by calculating the Cr3+ and O2- diffusion between various sites using the climbing-image Nudged-Elastic-Band (cNEB) approach. The activation energy ED (2.57 eV to 3.21 eV) obtained for the diffusion of Cr3+ is in good agreement with the experimental ED (2.46 eV). The calculated ED for O2- ranges from 2.21 eV to 3.65 eV, which is in agreement with experimental data. For each investigated diffusion pathway, frequencies calculated by finite difference methods are used to obtain jump frequencies using Transition State Theory (TST). Combining the pre-exponential factors with activation energies, the diffusion coefficients are calculated which are compared with experimental values.

Keywords: chromium oxide, magnetic moments, cation vacancies, anion vacancies, diffusion, activation energy, Nudged Elastic Band

2 ACS Paragon Plus Environment

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1. INTRODUCTION Chromia, Cr2O3, forms protective layers against corrosion on many industrial alloys

1,2.

Chromium is generally added to stainless steels in order to obtain on the surface a passive film rich in chromium oxide. For similar reasons, chromium is added to nickel-based alloys, thus coupling good mechanical properties and oxidation resistance in high temperature water. The role of such oxide layers is to limit the inward diffusion of oxygen and/or the outward diffusion of cations. The experimental investigations suggest that Cr2O3 scale growth is controlled by diffusion of elements (cations and anions). Several macroscopic models for growth kinetics of passive layer have been suggested

3-8.

They contained parameters adjusted from experiments.

The main limit of these models is that they consider the oxide layer as homogeneous and most of them do not allow the modeling of alloys. Atomistic simulation can be an interesting alternative to these macroscopic models 9-11. It allows taking into account the three dimensional structure of both alloy and oxide, and the presence of point defects. Such atomistic models, using empirical potentials, need local transport values obtained via ab initio methods like DFT in order to account for chemical and topological local environment. Therefore, it is necessary to understand the defect structure and corresponding transport properties in chromia. Bulk Cr2O3 is an anti-ferromagnetic semiconductor which crystallizes in the hexagonal corundum structure (space group R-3c) 12,13. The unit cell contains six formula units where two third of the octahedral sites are occupied by Cr and one third remains vacant as shown in Figure 1a. A number of studies has been performed for the defect properties 14-17 and diffusion in Cr2O3 16,18-24.

Experimental evidence

16

suggests that the intrinsic point defect equilibrium

predominates in chromia at high temperature. Chromia is an n-type semiconductor at low oxygen activities and a p-type semiconductor in near-atmospheric oxygen activities. Therefore, at reduced oxygen activities, point defects such as interstitial chromium ions or oxygen



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vacancies predominate, whereas at high oxygen activities, chromium vacancies or oxygen interstitial ions predominate. However, the ionic transport inside such material is disputed due to its strong dependence on the experimental conditions. Experimental investigations of ionic transport in chromia have been performed mainly in thermal oxidation conditions. Various factors affect the measured diffusion coefficient in chromia: types of alloys used, measurement conditions such as a wide range of temperature (700 to 1200 °C), oxygen partial pressures, impurity rates... 16. Therefore, divergent conclusions have been drawn in different studies for the diffusion of elements in chromia. Some studies have concluded that chromium diffusion is faster than oxygen diffusion 18,20,21,

whereas some others have concluded the opposite

22.

However, some experimental

investigations 19,23 have shown that diffusion coefficient values for both Cr and O diffusion in the bulk chromia are of the same order of magnitude. To our knowledge, there is only one theoretical investigation available where the growth of chromium passive layer in a Cr2O3 bulk has been investigated with DFT approach 25. Here Yu et al 25 have investigated vacancy diffusion pathways in Cr2O3 bulk along one direction, the z axis of the cell. The calculations were performed without taking into account the magnetism, with a pure DFT approach. However, a post-DFT formalism, such as DFT+U, is required to describe the strong intra-atomic electronic correlations in Cr2O3 which are modeled by adding an on-site Coulomb repulsion U to the DFT Hamiltonian 26,27.

In the present study, structural, electronic and magnetic properties of chromium and oxygen vacancies and of chromium Frenkel defects in bulk Cr2O3 are investigated theoretically with DFT+U approach. The diffusion processes of these point defects are modeled by using different representative pathways around the defect by applying the climbing-image Nudged-Elastic-


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Band (cNEB) method. Activation energies and frequencies are combined to compute full diffusion coefficients.

2. COMPUTATIONAL METHODS All calculations were performed from periodic calculations using the rotationally invariant DFT+U method, a post-DFT formalism developed by Dudarev et al.

28.

For Cr in Cr2O3, an

effective on-site coulomb interaction parameter U = 5 eV was used as defined in 26. This method was used as implemented in the plane-wave program VASP exchange-correlation functional

31

29,30.

The Perdew-Wang (PW91)

based on the GGA approximation was employed. The

projector-augmented wave (PAW) potentials 32,33 were used for the core electron representation. Whereas the valence electrons were represented by the energy cutoff Ecut = 520 eV as converged for the bulk Cr2O3 optimization in the present study. The Brillouin Zone integration was performed with a Monkhorst-Pack net 34. The convergence of bulk properties was checked with increasing k-point grids to a converged 3x3x1 on the Gamma-centered grid. DOS calculations were performed with 9x9x3 Gamma-centered grid. Bulk properties for the stoichiometric Cr2O3 were calculated with a single cell Cr12O18 and the defect calculations were performed with supercells ranging from 1x1x1, to 2x2x1 and 3x3x1. The transition state search for the diffusion processes was conducted with the climbingimage Nudged-Elastic-Band (cNEB) method 35 as implemented in VASP. Vibrational analysis calculations were performed to verify that initial and final states are true local minima and transition states are saddle point. No imaginary frequency was obtained for the local minima structures whereas one imaginary frequency was observed for the transition state structures. The calculated frequencies are used to evaluate the frequencies for each pathway using Transition State Theory (TST). Combining these values with the calculated diffusion energies leads to global diffusion coefficient D, according to the equation: 5 ACS Paragon Plus Environment


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D = D0 exp(

− ED ) kT

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(1)

Where ED is the diffusion energy, k denotes the Boltzmann constant, T is the temperature and D0 is the pre-exponential factor defined by

na 2 D0 = νd 6

(2)

Where n is the number of equivalent jumps, a is the distance between initial and final states and νd the jump frequency which can be obtained from real vibrational frequencies by 3N

Cν νd =

INIT j

j =1 3 N −1

∏ν TSj

(3)

j =1

Where νj INIT is the real vibrational frequencies of diffusing atom at initial state, and νjTS the real vibrational frequencies of diffusing atom at transitional state. Structural modeling was performed by using the graphical interface software ModelView 36.

3. RESULTS AND DISCUSSION 3.1 Stoichiometric Cr2O3. Prior to the study of the defect properties, we have investigated the defect-free Cr2O3 solid to test the method. The optimized lattice parameters a, c, shorter inter-Cr distance Cr-Cr (A), larger inter-Cr distance Cr-Cr (B), magnetic moment µ and optical band gap Eg are reported in Table 1. The most stable magnetic configuration is antiferromagnetic, with an inter-bilayer magnetic spin configuration along the c axis of (+-+-) 26,27,37,

as represented in Figure 1a with green (+) and grey (-) color for the Cr atoms. The

calculated magnetic moment (3.1 µB/atom) is underestimated compared to the experimental 6 ACS Paragon Plus Environment

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value (3.8 eV µB/atom 38). The calculated structural properties are in good agreement with the experiment

13

where a maximum deviation of 2.1 % is obtained for lattice parameter c. The

calculated values are close to those reported in other studies and obtained from the same level of theory

26,27,37.

Therefore, we conclude that the discrepancies between experimental and

calculated properties are inherent to the used DFT+U method 26,27. Cr2O3 is a semiconductor with an experimentally measured band gap of 3.4 eV

39.

The

electronic properties of the stoichiometric Cr2O3 are investigated by calculating the band structure and density of states (DOS). The band structure was computed along the path that contains the highest number of high symmetry points of the Brillouin Zone (BZ) inside hexagonal structure, namely Γ→M→K→ Γ→A→L→H→A.. The band structure is shown in Figure 1b. The top of the valence band (VB) and the bottom of the conduction band (CB) are located at M. The calculated minimum vertical transition (MVT) and minimum indirect transition (MIT) energies are given in Table 2. α-Cr2O3 has a direct band gap (Eg) of 2.8 eV along the M-M direction. The smallest indirect band gap is 3.1 eV along the M-K direction which is overall the third smallest band gap. The calculated total DOS (TDOS) and the projected DOS (PDOS) of O-2p and Cr-3d are shown in Figure 1c,d,e. The total DOS shows a semiconductive character with the presence of Cr-3d orbitals in both valence and conduction bands. The valence bands are contributed by Cr3d-t2g orbitals (not presented on the figure) whereas the conduction bands are contributed by Cr-3d-eg orbitals. Therefore, there are d→d transitions in the band gap of α-Cr2O3 which is a typical behavior of a Mott-Hubbard insulator, in agreement with previous results

26,40.

The

PDOS of oxygen (Figure 1e) shows that O-2p mainly contributes to the valence bands. It can be noted that Cr-3d and O-2p states contribute at the same level of energy in the valence band forming a hybridization which is typical of a charge-transfer insulator. From this hybridization and the above mentioned d→d transitions, chromia can be defined as an intermediate Mott-



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Hubbard – Charge transfer insulator, in agreement with both experimental 40 and theoretical 26 investigations.

3.2 Point Defects. Three types of point defects in Cr2O3 are studied, chromium vacancies, oxygen vacancies and chromium Frenkel defects. On the basis of the optimized structural parameters for the perfect crystals, 1x1x1, 2x2x1 and 3x3x1 supercells are constructed for defect calculations. The defective system is considered as a new crystal with an artificially introduced point defect periodicity. The defect concentration ranges from 3.45 to 0.37%, which is much larger than the real defect concentration (10-6% in the case of high PO2 at 1100°C 14). In the present section, the formation energy of a defect in Cr2O3, the effect of relaxation for this type of defect and the influence of this defect on the electronic and magnetic properties are studied. For each of the supercells, the defect formation energy of a defect type X has been calculated according to the equation: EV(X) = ET(Cr2O3) - ET(Cr2O3)(defect) + µ(X)

(4)

Here ET(Cr2O3) and ET(Cr2O3)(defect) denote the total energy of the supercell without and with point defect respectively, and µ(X) is the chemical potential of a free X atom (X = Cr or O). µ(Cr) is calculated from the energy of a single Cr atom inside an optimized pure Cr bulk cell. µ(O) is calculated from the energy of an atom in a single O2 molecule. µ is equal to 0 in the case of Frenkel defect, as the system stays stoichiometric. Within DFT+U method, total energies are dependent on the U value. However, as our calculated data are based on differences in energies instead of absolute values, this energy dependence should not be critical as the general trends will be unaffected by the U value.


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3.2.1 Cr Vacancy. The vacancy was created by removing one Cr atom from a supercell (1x1x1, 2x2x1 or 3x3x1), keeping the system neutral. All Cr atom sites are equivalent. Therefore the importance of the location of the Cr atom inside the cell is negligible. In Table 3, the calculated chromium vacancy formation energies EV(Cr) are presented. The calculated EV(Cr) ranges between 4.84 and 5.04 eV. It should be noted that the convergence of EV(Cr) with respect to the supercell size is slow. However, due to limitation of computer resources, we were not able to calculate supercells larger than 3x3x1, which would probably help find proper convergence in the calculated defect formation energy values. The shortest distance between two defects is 10.0 Å in the case of the 2x2x1 supercell which is assumed to be sufficient to model isolated defects. The structural relaxation effects are investigated by measuring the changes of distances of the relaxed atoms from the Cr defect position. Calculated results are shown in Table 4. In nondefective Cr2O3, the Cr atom is surrounded by six oxygen atoms as the first nearest neighbors (1-NN). After relaxation, the 1-NN oxygen atoms show an outward relaxation from the vacancy of 5.7%. This is consistent because the electrostatic attraction by the Cr3+ cation is missing. The removal of one neutral Cr atom creates three holes in the valence band. There are four chromium atoms as 2-NN. Three of them show a strong inward relaxation of -5.5% and the fourth one shows an even larger relaxation of -7.4%. The latest one is separated from the vacancy by a compact plane of O atoms, unlike the three other ones which are located in the same Cr bi-layer as the vacancy and, therefore, are equivalent. Because of the reduced electrostatic repulsion, the 2-NN Cr atoms tend to move closer to the vacancy. Other atoms in next coordination shells (3-NN) show only small relaxation. We conclude that the relaxation is mainly restricted to the first nearest and the second nearest neighbor atoms, typical of a polaronic distortion. The DOS for a defective 2x2x1 supercell is shown in Figure 2a. It can be seen that the neutral Cr vacancy introduces an extra unoccupied level roughly 0.3 eV above the Fermi level (EF)



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(marked by an arrow in Figure 2a). Transitions from the valence band to this level occur at lower energy than the band gap transitions. This level is mainly composed of oxygen p orbitals from atoms surrounding the vacancy site. It is found that the p orbitals of three of the six nearest oxygen atoms have much larger contributions than those of the other atoms. The removal of a neutral Cr atom creates holes in the valence band. Three of the surrounding oxygen atoms, which were formally O2- in stoichiometric Cr2O3, become O-. The observed defect level can be seen as an acceptor level. The magnetic configuration of the defective supercell shows a change with respect to the magnetic ground state of the non-defective supercell, especially in the vicinity of the vacancy. The magnetic configuration remains antiferromagnetic but magnetic moments of the first nearest neighbors of the vacancy are lower: the 6 O atoms around the vacancy have µ = -0.3 µB instead of 0 µB. Atoms located further have no lowering of their magnetic moment. The total lowering of the magnetic moment on the 1-NN O atoms corresponds to the localized compensation for the charge loss caused by the removal of Cr.

3.2.2 Chromium Frenkel Defect. One third of the octahedral sites inside the oxygen hexagonal structure are unoccupied in the non-defective chromia (see Figure 1a). To create a Frenkel defect, one Cr atom was moved from a regular site to one of these neighboring unoccupied octahedral sites. Depending on the location of the Cr vacancy left by the interstitial, two types of Frenkel defect are possible (Figure 3a): (i) the vacancy and the interstitial are on the same Cr bi-layer (configuration A) or (ii) they are separated by an O compact plane (configuration B). In both cases, the point defect is located in the second nearest neighbor position with respect to the unoccupied octahedral site. The Frenkel formation energy, EFr(Cr), was calculated as the energy needed to move a Cr3+ ion from its regular lattice site to an interstitial site. In Table 3, the calculated EFr(Cr) values


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are presented for the two different configurations with fully relaxed supercells. Our investigation shows that, Frenkel formation is easier for the ‘configuration A’ compared to ‘configuration B’. The most favorable ‘configuration A’ contains the vacancy and the interstitial in the same Cr bi-layer, leaving the same number of Cr atoms per bi-layers. On the other hand, the ‘configuration B’ brings a new Cr atom inside the previously-completed Cr bilayer, leaving another bi-layer with one less Cr atom. Our calculated EFr(Cr) for ‘configuration A’ is ≈ 2.5 eV, which is much smaller than the Cr vacancy formation energy (≈ 5.0 eV). Therefore, we conclude that interstitial chromium vacancy is easier to form than chromium vacancy in chromia. However, achieving convergence is slow for EFr(Cr) with respect to the supercell size, similar to the Ev(Cr). The similarity of the trends obtained for Ev(Cr) and EFr(Cr) is not surprising, since both defects involve the formation of an empty Cr lattice site. The two defect types can be regarded as extreme cases of real lattice defects, where the relocated Cr is close to the vacancy (Frenkel defect) or at infinite distance (hole vacancy). Structural relaxations around the Cr Frenkel defect in configuration A were investigated by measuring the distances between the newly-created vacancy and its neighbors as well as between the newly-occupied Cr site and its neighbors. The results are shown in Table 5. In the vacancy’s vicinity, it is observed that the oxygen atoms in the 1-NN show an outward relaxation by +5.0 to +7.0 %. This is reasonable as the 1-NN oxygen atoms are less attracted toward the position of the vacancy due to the absence of the electrostatic attraction of the removed chromium atom. On the contrary, the 2-NN Cr atoms are attracted by the vacancy due to the absence of the electrostatic repulsion of due to the removed chromium atom. The changes in the neighborhood around the vacancy follow the same trend as around the single Cr vacancy described above. Around the interstitial, the relaxations occur in the opposite direction. The 2 Cr atoms located on the two bilayers above and below the interstitial move significantly (outward displacement of +13.5%-+17.9%). The 6 oxygen atoms in the coordination



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octahedron around the interstitial show an inward relaxation. Two oxygen have a slight displacement (-1.3%) because they are not in the coordination octahedron around the vacancy. The four others have a larger inward relaxation (-4.4%--8.0%), due to the repulsive effect of the vacancy and the attractive effect of the interstitial. Two of the 3-NN Cr atoms show a strong outward displacement (+4.0%) corresponding to their inward displacement to the vacancy. Three other Cr atoms show only a slight relaxation (-0.5%) because they are located on the opposite side to the vacancy. The DOS for a defective 2x2x1 supercell is shown in Figure 3b. It can be seen that the Cr Frenkel defect introduces an extra occupied donor level, as a spin up state, roughly 0.5 eV below the Fermi level (EF), which is marked by an arrow (Figure 3b). Transitions from this level to the conduction band occur at lower energy (2.1eV) than the band gap transitions. The occupied defect level consists mainly of chromium states, contributed mainly by the 3d orbital of the Cr interstitial atom. The presence of the level as a spin up state is due to the chosen Cr atom and its magnetic moment (spin down), if a magnetic spin up configuration would be chosen, the occupied defect level would have been a spin down state. The magnetic configuration remains antiferromagnetic. The interstitial Cr atom keeps the same magnetic moment as in its normal site.

3.2.3 Oxygen Vacancy. As for the chromium vacancy, the oxygen defective supercell was modeled by removing a single O atom from a regular lattice site. The oxygen vacancy formation energy EV(O) was calculated with Equation 4 for different supercells with increasing size. The calculated values of EV(O) are presented in Table 3. The formation energy of oxygen vacancy is only slightly higher than Cr vacancy formation energy (which may be related to the fact that Cr2O3 has no specific dominant native defect in near equilibrium growth conditions and the oxygen partial pressure is the major factor governing the type of defect 2,15 ). It is much higher than the Cr Frenkel formation energy. 12 ACS Paragon Plus Environment

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The structural relaxation effects of the oxygen vacancy were investigated by measuring the changes of distances of the relaxed atoms from the defect position. The observed changes are shown in Table 6. Compared to the Cr vacancy and the Cr Frenkel defect, the structural relaxation is weaker around the O vacancy, with a mean displacement of 0.03 Å. Cr atoms in the first coordination cell (1-NN) show a very small outward displacement for half of them (+1.5%) and smaller inward displacement for the other half (-0.9%). The 2-NN O atoms show a relaxation of -1.04% to +0.2%, indicating that the positions of the oxygen atoms are almost unchanged. The magnetic configuration is slightly distorted around the vacancy. There is an enhancement of absolute value of the magnetic moment of the 1-NN Cr atoms around the O vacancy by 0.2

µB (|µ | = 3.3 µB instead of 3.1 µB). It shows that the 2 unpaired electrons due to the O vacancy are localized mainly on the Cr atoms surrounding the vacancy. The DOS for a defective 2x2x1 supercell is shown in Figure 4a. It can be seen that the O vacancy introduces no extra unoccupied level in the band gap, unlike the other defects. It has already been observed in other materials that neutral oxygen defects could induce such a situation, due to the difference of electronegativity between O and transition metal 41.

3.3 Diffusion of ions in Cr2O3. We have studied the diffusion of both Cr3+ and O2- species considering various possible diffusion pathways in bulk Cr2O3, by using a 2x2x1 supercell. The diffusion of a single Cr cation is accomplished via either chromium vacancies or a chromium Frenkel defect whereas O diffusion occurs via oxygen vacancies. In each diffusion case, we proceeded by moving a neighboring atom to a pre-existent empty site (Frenkel defect) or a vacancy created by removing a Cr or an O atom.



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3.3.1 Diffusion of Cr via a chromium Frenkel defect. One third of cation sites in the chromia structure are normally vacant and can serve either as a final position for a migrating neighboring cation or as a metastable intermediate site along a far-stretched diffusion path through a multiple step process. We have explored chromium diffusion to the nearest unoccupied interstitial position, thus creating a Cr Frenkel defect (Figure 5a). Due to symmetry equivalence, only two pathways have been considered. First, Cr3+ can migrate within the same Cr bi-layer in the xy plane towards the unoccupied octahedral site, i. e. the pathway A→B, at a distance of 2.9 Å (Figure 5b). Another possibility for Cr3+ diffusion is in the z direction, through the compact O layer, i. e. the pathway A→C, at a distance of 2.7 Å (Figure 5c). In Table 7, the calculated diffusion energies (ED) are reported. Our calculated ED value in the xy plane is 3.01 eV (A→B), whereas ED along the z direction is 5.98 eV (A→C). This shows that the diffusion of the Cr3+ ion along the z direction (A→C) would be more difficult than in the xy plane (A→B). In the two cases, the final position is less stable than the initial position (Figure 5d) illustrating the fact that the unoccupied octahedral site is the least stable site for Cr. In addition the barrier for returning to the initial position is very low, limiting the resident time in the interstitial position. These two factors could explain why an overpopulated cation bilayer is not favored over a stoichiometric cation bi-layer. As the two pathways start with the same initial configuration, the difference in the diffusion barrier may be related to the difference between the structures of the transition states (TS). The structural analysis reveals distinct features inside the first and second neighbors of the diffusing atom for the two diffusion pathways (as shown on Figure 5b,c). Chromium diffusion in the xy plane (A→B) occurs through the vertex of the coordination polyhedron as shown in Figure 5b. TS on this pathway consists of 4 oxygen atoms (Cr–O distance ranging between 1.8 – 2.2 Å) as the first nearest neighbors (1-NN) and 2 chromium atoms (Cr–Cr distance ranging between


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2.5 – 2.6 Å) as second nearest neighbors (2-NN). The diffusion along the z direction occurs through the facet of the same polyhedron (A→C), as shown on Figure 5c. TS structure along A→C contains 3 oxygen atoms (1.9 Å) and 7 chromium atoms (3.1-3.2 Å). Due to the higher number of Cr atoms in the nearest neighbors around TS, there is a larger electrostatic repulsion in the case of A→C pathway compared to the A→B pathway. Therefore the activation barrier is almost twice larger for A→C than for A→B.

3.3.2 Diffusion of Cr via a Cr vacancy. Another possibility of cation diffusion is explored by the diffusion of Cr via a vacancy mechanism. Depending on the location of Cr ions in the vicinity of this vacancy, there are various distinct diffusion pathways as depicted in Figure 6a. Cr can migrate inside the same Cr bi-layer in the xy plane towards any of the three equivalent neighboring Cr atoms, at a distance of 2.9 Å. Only one of these three pathways (A→B) is represented on Figure 6b. Two other possibilities of Cr3+ diffusion are in the z direction such as (a) diffusion through the compact O layer (A→C: distance 2.7 Å, Figure 6c) and (b) Cr3+ diffusion through two compact O layers via an octahedral unoccupied site (A→D: distance 4.9 Å, Figure 6d). The calculated diffusion energies (ED) are reported in Table 7. The calculated value of ED in the xy plane is 2.73 eV (A→B), whereas ED along the z direction is either 2.57 eV for A→D or 3.21 eV for A→C. It can be noted that the diffusion of the Cr3+ ion along the z direction through the unoccupied octahedral site (A→D) would be easier than along the two other pathways. Even if the given diffusion energy is rather small, the Cr3+ ion must diffuse through an O compact layer two times in order to complete the full pathway. However, the total energy of the intermediate state is only 0.2 eV above those of the initial state (Figure 6e), favoring a significant resident time in this metastable position and then the opportunity of a second jump towards the end of the path.



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The structural analysis of TS reveals distinct features inside the first and second neighbors of the diffusing atom for the various diffusion pathways. Chromium diffusion along the xy plane (A→B) occurs through the vertex of the coordination polyhedron as shown in Figure 6b. TS for the diffusion along A→B consists of 4 oxygen atoms (ranging between 1.8 – 1.9 Å) and 2 chromium atoms (at 2.9 Å). On the other hand, diffusion along the z direction occurs through the facet of the same polyhedron (A→C and A→D). The TS for the diffusion pathway along A→C contains 3 oxygen (1.8 Å) and 6 chromium (3.1 Å) atoms and the TS along the A→D contains 3 oxygen (1.8 Å) and 3 chromium (3.1 Å) atoms as the nearest neighbors. The TSs of the A→C and A→D pathways have the same number of O atoms at the same distance, consequently similar steric effects due to 1-NN. So the higher barrier for path A→C compared to A→D is mainly related to the fact that A→C TS has twice the number of Cr atoms as 2-NN than A→D TS, therefore higher electrostatic repulsion. However path A→B has less 2-NN Cr atoms than A→D but higher 1-NN O atoms. In this case, the decrease in electrostatic repulsion (less Cr atoms) is compensated by a strong steric effect (more O atoms). Therefore, the activation energy for the A→B pathway (ED = 2.7 eV) is slightly larger than that for the A→D pathway (2.57 eV).

3.3.3 Diffusion of O via an oxygen vacancy. With 12 oxygen atoms surrounding the newlycreated vacancy, various distinct oxygen diffusion pathways are possible, as depicted in Figure 7. O2- can migrate within the same O compact layer in the xy plane towards two different sites, A→B and A→C at distances of 2.8 and 2.6 Å, respectively. Other possibilities of O2- diffusion are in the z direction, through a Cr bi-layer such as (a) diffusion to a distance of 2.7 Å (A→D) and (b) diffusion to a distance of 3.0 Å (A→E). Oxygen diffusion along the xy plane occurs either through the vertex of the coordination polyhedron (A→B) as shown in Figure 7b or through the facet of the same coordination


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polyhedron (A→C, Figure 7c). The diffusion along the z direction occurs either through the facet of the same polyhedron (A→D). Or it can occur through the vertex (A→E). In Table 7, the calculated diffusion energies (ED) are reported. The calculated values of ED in the xy plane range between 2.21 eV (A →C) and 3.22 eV (A→B), whereas ED along the z direction ranges between 2.35 eV (A→D) and 3.65 eV (A→E). This shows that the diffusion of the O2- ion through facets of coordination polyhedron around the vacancy site (A→C or A→D) would be easier than those through vertices (A→B or A→E). Also oxygen diffusion pathways are symmetrical as shown on Figure 7f.

3.3.4 Comparison with experimental data. Using our calculated ED values and vibrations calculations at initial and TS states (Equation 3), we applied Transition State Theory in order to calculate the pre-exponential factors D0 (Equation 2) and global diffusion coefficients DM (Equation 1) for Cr Frenkel defects, O vacancies and Cr vacancies. Calculated and experimental values of migration energies ED, pre-exponential factors D0 and global diffusion coefficients DM for O and Cr vacancies 14,42, measured at 900°C are reported in Table 7. Specific experimental values for diffusion via a Cr Frenkel defect mechanism are not available and the values given for Cr diffusion are certainly more related to a Cr vacancy mechanism due to p-type character of the involved Cr2O3 material in these studies. Our calculated ED and D are in close agreement with experimental values (Table 7). From these values, it can be considered that oxygen diffusion is faster than Cr diffusion in Cr2O3 which is in agreement with a part of the available contradictory experimental data 22. Here it should be mentioned that the orders of magnitude of calculated DM are too close to conclude on the most suitable pathway. Experiments give a statistical value for DM, obtained from the probable diffusion of several atoms following different pathways while DFT calculations provide values only for specific pathways. To go beyond this first comparison, the



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data obtained from DFT calculations on the individual pathways can be combined through a kinetic Monte Carlo simulation to produce a global diffusion coefficient directly comparable to those obtained from tracer experiments 43. This study is underway.

4. CONCLUSION The structural, energetic, electronic, magnetic, defective and transport properties of bulk Cr2O3 have been investigated by first principles DFT+U approach. Three different point defects, Cr vacancy, Cr Frenkel defect (composed by an interstitial Cr atom and a Cr vacancy) and O vacancy are considered. All of these defects are reported to influence drastically electronic properties of Cr2O3 and are involved in diffusion processes in the growth of passive film. According to our calculated defect formation energy, the stability of defects follows the order: Cr Frenkel defect > Cr vacancy > O vacancy, in agreement with the p-conductor type of Cr2O3 at low temperatures. Relaxation was found to occur only on the first and the second nearestneighbors. On an electronic point of view, Cr Frenkel defect and Cr vacancy defect add an extra level inside the band gap. In the case of the Cr Frenkel defect, the level added by the interstitial Cr atom is donor; in the case of the Cr vacancy defect, the level added by O is acceptor. O vacancies do not induce any level in the band gap. Defects also influence the magnetic moments on atoms around them, especially on the first and the second nearest-neighbors. The diffusion of Cr and O has been investigated via a Frenkel defect, a Cr vacancy and an O vacancy. Several pathways have been considered and the transition states have been analyzed in order to understand the differences between diffusion energies. Two effects are responsible for higher barrier energy: electrostatic repulsion for Cr Frenkel diffusion, and steric effects for


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O diffusion. Relative balance between these two effects depends on the considered Cr vacancy pathway. The calculated diffusion energy reveals that O diffusion (ED = 2.35 eV) is faster than Cr diffusion (ED = 2.57 eV) which is in agreement with experimental data. The calculated diffusion coefficients for all considered diffusion pathways show that oxygen diffusion is faster than chromium diffusion, in agreement with some experimental studies 22.



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


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Table 1. Calculated lattice parameters a and c (Å), bond distances (Å), band gap Eg (eV) and magnetic moment µ (µB/atom) for bulk Cr2O3 and comparison with previously calculated values and experimental data. Calculated (this study) Calculated 26

Experimental 13

a (Å)

5.074

5.073

4.951

c (Å)

13.850

13.839

13.566

Cr-Cr(A) (Å)

2.717

2.723

2.623

Cr-Cr(B) (Å)

4.027

4.196

4.085

Eg (eV)

2.8

2.6

3.4

µ (µB/atom)

3.1

3.01

3.8



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Table 2. Band gap energies Eg (in eV) depending on the transition type and the concerned high symmetry point in the Brillouin zone. MVT means Minimum Vertical Transition and MIV Minimum Indirect Transition. MVT

Eg (eV)

MIV

Eg (eV)

Γ→Γ

3.26

Γ→M

3.06

M→M

2.82

M→K

3.06

K→K

3.11

K→Γ

3.08

A→A

3.25

A→L

3.10

L→L

3.00

L→H

3.22

H→H

3.23

H→A

3.16

Γ→A

3.28


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Table 3. Defect formation energies for Cr vacancy EV(Cr), Cr Frenkel defect (configuration A) EFr(Cr) and O vacancy EV(O) (in eV) as function of defect concentration c (% of atoms). EFr(Cr) given in brackets represents the Frenkel defect formation energy for the configuration B. Supercell size

c

EV(Cr) (eV)

EFr(Cr) (eV)

EV(O) (eV)

1x1x1

3.4

5.04

3.45 (3.09)

5.21

2x2x1

0.8

4.98

2.49 (4.99)

5.23

3x3x1

0.4

4.84

2.36 (5.07)

5.12

Table 4. Distances r (Å) of neighboring atoms from the defect and changes of the distance r (%) for relaxed atoms around the Cr vacancy (2x2x1 supercell). Between brackets, the number of equivalent neighbors. Atom

(nb

equivalent atoms)

of Type

of Unrelaxed

neighborhood

O (3)

(Å)

Relaxed

∆r (%)

(Å)

1.92

2.03

+5.7

O (3)

2.24

2.11

+5.7

Cr (1)

2.52

2.33

-7.4

Cr (3)

2.94

2.78

-5.5

O (3)

3.50

3.49

-0.5

3.61

3.60

-0.2

1-NN

2-NN

3-NN Cr (6)



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Table 5. Distances r (Å) of neighboring atoms around a Cr Frenkel defect (both around the Cr interstitial and the vacancy). Changes of the distance ∆r (%) after relaxation (2x2x1 supercell). Between brackets, the number of equivalent neighbors. Crint denotes the newly created Cr interstitial. Atom (nb of Type equivalent

neighborhood

of Unrelaxed

Relaxed

∆r (%)

(Å)

(Å)

2.01

2.10

+5.0

O (3)

2.05

2.20

+7.0

Cr (1)

2.72

2.56

-5.3

2.94

2.94

+0.0

Cr (3)

2.96

2.83

-4.0

Cr (1)

2.10

2.38

+13.5

Cr (1)

2.10

2.48

+17.9

O (2)

2.11

2.08

-1.3

2.11

2.02

-4.4

O (2)

2.11

1.94

-8.0

Cr (2)

2.94

3.04

+4.0

2.94

2.91

-0.5

atoms) Vacancy

O (3) 1-NN

Crint (1)

Interstitial

2-NN

1-NN

O (2)

2-NN

3-NN Cr (3)


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Table 6. Distances r (Å) of neighboring atoms from the O vacancy and changes of the distance ∆r (%) for relaxed atoms around the O vacancy (3x3x1 supercell). Between brackets, the number of equivalent neighbors. Atom

(nb

of Type of neighborhood

∆r (%)

Unrelaxed

Relaxed

(Å)

(Å)

1.91

1.89

-0.9

Cr (2)

2.23

2.26

+1.5

O (2)

2.78

2.74

-1.4

2.84

2.83

-0.4

3.07

3.07

+0.2

equivalent atoms)

Cr (2) 1-NN

O (2) O (2)

2-NN



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Table 7. Diffusion distances dD (Å), diffusion energies ED (eV), pre-exponential factor D0 (cm2.s-1) and global diffusion coefficient DM (cm2.s-1) for each of the pathways involving a Cr vacancy, a O vacancy or a Cr interstitial at 1173K. dD (Å)

ED (eV)

D0 (cm2.s-1)

DM (cm2.s-1)

Frenkel A-B

2.9

3.01

5.0x10-4

5.4x10-17

A-C

2.7

5.98

6.2x10-4

1.3x10-29

A-B

2.9

2.73

9.4x10-4

1.7x10-15

A-C

2.7

3.21

5.8x10-4

9.4x10-18

A-D

4.9

2.57

2.4x10-4

2.2x10-15

Exp.14,42

-

2.9

2.8x10-2

1x10-14 - 2x10-18

A-B

2.8

3.22

1.3x10-3

3.8x10-17

A-C

2.6

2.21

5.8x10-4

8.0x10-15

A-D

2.7

2.35

3.4x10-3

5.6x10-13

A-E

3.0

3.65

2.0x10-3

8.6x10-19

Exp.14,42

-

2.5

5.4x10-4

1x10-15 - 3x10-19

Pathway Cr defect

Cr vacancy

O vacancy


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Figure 1. (a) Structure of α-Cr2O3. The green and gray disks are chromium atoms, with respectively up and down magnetic spins. Red disks are oxygen atoms. Dashed circles are some examples of unoccupied octahedral sites. (b) Band Structure of bulk Cr2O3 along

Γ→M→K→ Γ→A→L→H→A. (c) Density of states (DOS) of nondefective bulk Cr2O3. (d) Cr3d projected DOS. (e) O-2p projected DOS.

(a)

Α

Β

(b)



40

Density of States (a.u.)

(c)

20

0

-20

-40 -6

-4

-2

0

2

4

6

Energy (eV)

4


(d)

2

0

-2

-4 -6

-4

-2

0 Energy (eV)

2

4

6

-6

-4

-2

0

2

4

6

4

(e) Densisty of States (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 28 of 46

2

0

-2

-4 Energy (eV)


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Figure 2. (a) Density of states (DOS) of bulk Cr2O3 with Cr vacancies. (b) Cr-3d projected DOS. (c) O-2p projected DOS.

400


(a)

200

0

-200

-400 -6

-4

-2

0

2

4

2

4

Energy (eV)

4

(b)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2

0

-2

-4 -6

-4

-2

0 Energy (eV)



4

(c)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 46

2

0

-2

-4 -6

-4

-2

0

2

4

Energy (eV)


Page 31 of 46

Figure 3. (a) Location of Cr vacancy in Frenkel defect in the two A and B configurations. The arrows show the two possibilities for the interstitial Cr (denoted by a doted line circle). (b) Density of states (DOS) of bulk Cr2O3 with Cr Frenkel defect. (c) Cr-3d projected DOS. (d) O-2p projected DOS. (a) A

B

(b)

Density of state (a.u.)

150

0

-150

-300

-450 -8

-6

-4

-8

-6

-4

-2 0 Energy (eV)

2

4

2

4

3

(c)

2 Density of state (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1 0 -1 -2 -3 -2

0

Energy (eV)



3

(d)

2 Density of state (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 32 of 46

1 0 -1 -2 -3 -8

-6

-4

-2

0

2

4

Energy (eV)


Page 33 of 46

Figure 4. (a) Density of states (DOS) of bulk Cr2O3 with O vacancy. (b) Cr-3d projected DOS. (c) O-2p projected DOS. 400


(a)

200

0

-200

-400 -8

-6

-4

-2

0

2

4

Energy (eV)

3

(b) Density of States (a.u.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1

-1

-3 -8

-6

-4

-2

0

2

4

Energy (eV)



3

(c)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 34 of 46

1

-1

-3 -8

-6

-4

-2

0

2

4

Energy (eV)


Page 35 of 46

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Figure 5. (a) Diffusion pathways in the case of Cr Frenkel defect. A is the center of the blue octahedron around the unoccupied interstitial site. B (yellow and green, both are equivalent) and C (blue) are the two diffusing atoms. (b) A→B diffusion pathway. Diffusion occurs along the black line from the initial position of the diffusing atom to the centre of the octahedron. (c) A→C pathway. (d) Frenkel Cr diffusion energies (in eV) versus fractional distance between initial and final positions for each diffusion pathways.

(a)

A

(b)

A

(c)

A→B

A→C



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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(d)

C B A


Page 37 of 46

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Figure 6. (a) Diffusion on pathways in the case of Cr vacancy. A is the center of the blue octahedron around the Cr vacancy. B (yellow), C (green) and D (grey) are the four diffusing atoms. (b) A→B diffusion pathway. Diffusion occurs along the black line from the initial position of the diffusing atom to the center of the octahedron. (c) A→C pathway. (d) A→D pathway. (e) Cr diffusion energies (in eV) versus fractional distance between initial and final positions for each diffusion pathways. (a)

(b)

A→B

(c)

(d)

A→C

A→D



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 38 of 46

(e)


Page 39 of 46

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Figure 7. (a) Diffusion pathways in the case of O vacancy. A (red, inside the Cr tetrahedron) is the center of the tetrahedron around the O vacancy. B (green), C (light blue), D (yellow) and E (dark blue) are the four diffusing atoms considered in the study. (b) A →B diffusion pathway. Diffusion occurs along the black line from the initial position of the diffusing atom to the centre of the octahedron. (c) A→C pathway. (d) A→D pathway. (e) A→E pathway. (f) O diffusion energies (in eV) versus fractional distance between initial and final positions for each diffusion pathways. (a)

C

(b)

B

(c)

C

B A→B

(d)

A→C

(e)

D

E

A→D

A→E



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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(f)


Page 41 of 46

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REFERENCES (1) Kofstad, P. High Temperature Corrosion; Elsevier Applied Science Publishers Ltd.: London, U.K., 1988. (2) Marcus, P. Corrosion Mechanisms in Theory and Practice. 3rd edition; CRC Press: Boca Raton, FL, 2012. (3) Wagner, C. Theory of the Tarnishing Process. Z. Phys. Chem., Abt. B 1933, B21, 25-41 (4) Cabrera, N.; Mott, N. F. Theory of the Oxidation of Metals. Rep. Prog. Phys. 1949, 12, 163-184 (5) Fehlner, F. P.; Mott, N. F. Low-Temperature Oxidation. Oxid. Met. 1970, 2, 59-99 (6) Macdonald, D. D. Passivity – The Key to our Metals-Based Civilization. Pure Appl. Chem. 1999, 71, 951-978 (7) Seyeux, A.; Maurice, V.; Marcus, P. Oxide Film Growth Kinetics on Metals and Alloys I. Physical Model. J. Electrochem. Soc., 2013, 160, C189–C196. (8) Leistner, K.; Toulemonde, C.; Diawara, B.; Seyeux, A.; Marcus, P. Oxide Film Growth Kinetics on Metals and Alloys II. Numerical Simulation of Transient Behavior. J. Electrochem. Soc., 2013, 160, C197-C205. (9) Legrand, M.; Diawara, B.; Legendre, J. J.; Marcus, P. Three-Dimensional Modelling of Selective Dissolution and Passivation of Iron-Chromium Alloys. Corros. Sci.2002, 44, 773– 790. (10) Diawara, B.; Legrand, M.; Legendre, J.J.; Marcus, P. Use of Quantum Chemistry Results in 3D Modeling of Corrosion of Iron-Chromium Alloys. J. Electrochem. Soc. 2004, 151, B172B178. 41 ACS Paragon Plus Environment


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(11) Diawara, B.; Beh, Y.A.; Marcus, P. Nucleation and Growth of Oxide Layers on Stainless Steels (FeCr) Using a Virtual Oxide Layer Model. J. Phys. Chem. C 2010, 114, 19299-19307. (12) Pauling, L.; Hendricks, S. B. The Crystal Structures of Hematite and Corundum. J. Am. Chem. Soc., 1925, 47, 781–790. (13) Finger, L. W.; Hazen, R. M. Crystal Structure and Isothermal Compression of Fe2O3, Cr2O3, and V2O3 to 50 kbars. J. Appl. Phys., 1980, 51, 5362–5367. (14) Sabioni, A. C. S.; Lesage, B.; Huntz, A. M.; Pivin, J. C.; Monty, C. Self-Diffusion in Cr2O3 I. Chromium diffusion in Single Crystals. Philos. Mag. A, 1992, 66, 333–350. (15) Sabioni, A. C. S.; Lesage, B.; Huntz, A. M.; Millot, F.; Monty, C. Self-Diffusion in Cr2O3 II. Oxygen Diffusion in Single Crystals. Philos. Mag. A, 1992, 66, 351–360. (16) Holt, A.; Kofstad, P. Electrical Conductivity and Defect Structure of Cr2O3. I. High Temperatures ( > 1000°C). Solid State Ionics, 1994, 69, 127–136. (17) Atkinson, K. J. W.; Grimes, R. W.; Levy, M. R.; Coull, Z. L.; English, T. Accommodation of Impurities in alpha-Al2O3, alpha-Cr2O3 and alpha-Fe2O3. J. Eur. Ceram. Soc., 2003, 23, 3059–3070. (18) Tsai, S. C.; Huntz, A. M.; Dolin, C. Diffusion of 18O in Massive Cr2O3 and in Cr2O3 Scales at 900°C and its Relation to the Oxidation Kinetics of Chromia Forming Alloys. Oxid. Met., 1995, 43, 581–596. (19) Tsai, S. C.; Huntz, A. M.; Dolin, C. Growth Mechanism of Cr2O3 Scales: Oxygen and Chromium Diffusion, Oxidation Kinetics and Effect of Yttrium. J. Mater. Sci. Eng. A, 1996, 212, 6–13.


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Structural, Magnetic, Electronic, Defect, and Diffusion Properties of

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