Optical Absorption and Band Gap Reduction in (Fe1âxCrx) - American

Article pubs.acs.org/JPCC

Optical Absorption and Band Gap Reduction in (Fe1−xCrx)2O3 Solid Solutions: A First-Principles Study Yong Wang,† Kenneth Lopata,‡ Scott A. Chambers,† Niranjan Govind,‡ and Peter V. Sushko*,§ †

Fundamental and Computational Sciences Directorate, Pacific Northwest National Laboratory, Richland, Washington 99352, United States ‡ Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratory, Richland, Washington 99352, United States § Department of Physics and Astronomy and London Centre for Nanotechnology, University College London, Gower Street, London WC1E 6BT, United Kingdom S Supporting Information *

ABSTRACT: We provide a detailed theoretical analysis of the character of optical transitions and band gap reduction in (Fe1−xCrx)2O3 solid solutions using extensive periodic model and embedded cluster calculations. Time-dependent density functional theory is used to calculate and assign optical absorption bands for x = 0.0, 0.5, and 1.0 and photon energies up to 5 eV. Consistent with recent experimental data, a band gap reduction of as much as 0.7 eV with respect to that of pure α-Fe2O3 is found. This result is attributed predominantly to two effects: (i) the higher valence band edge for x ≈ 0.5, as compared to those in pure α-Fe2O3 and αCr2O3, and (ii) the onset of Cr → Fe d−d excitations in the solid solutions. Broadening of the valence band due to hybridization of O 2p with Fe and Cr 3d states also contributes to band gap reduction.

1. INTRODUCTION The increasing global demand for energy reinforces the need for sustainable energy generation. Harvesting solar energy in, for example, photochemical production of hydrogen and photovoltaics is a promising path forward. However, for this approach to be of practical use, greater efficiency in capturing sunlight needs to be achieved.1 In particular, materials that have an optical band gap of about 1.3−1.4 eV and are at the same time inexpensive, environmentally friendly, and in possession of tunable electronic properties are highly desirable.2 Hematite (αFe2O3) has attracted interest in this regard because it is abundant in the Earth’s crust, nontoxic, stable, easily synthesized, and inexpensive. Hematite is a charge transfer insulator,3 has a band gap of ∼2.0 eV,4 absorbs ∼40% of the solar spectrum,5 and conducts when doped with Ti.6 These properties make hematite very promising for solar energy conversion applications. Isostructural and isovalent α-Fe2O3 and α-Cr2O3 and their solid solutions are the subject of active experimental and theoretical research.2,4−12 It is expected that the photoabsorption characteristics of α-Fe2O3−Cr2O3 alloys can be tuned without dramatically affecting structural and redox properties. A similar approach has been exploited to shift the optical absorption of MgO nanoparticles from the ultraviolet region to the visible region of the electromagnetic spectrum by doping them with CaO and, thus, lowering their effective optical gap.13 A recent experimental study reported that the band gap of (Fe1−xCrx)2O3 mixtures can be reduced to ∼1.6 eV, which is smaller than those of α-Cr2O3 (3.4 eV) and α-Fe2O3 (2.2 eV).9 Interestingly, the minimum band gap exhibited by © 2013 American Chemical Society

(Fe1−xCrx)2O3 is also close to the optimum value for highly efficient solar optical absorption.9 Previously, the band gap narrowing observed in (Fe1−xCrx)2O3 was explained in terms of band alignment in the α-Fe2O3/α-Cr2O3 heterostructure, where the valence band maximum (VBM) of α-Cr2O3 is higher than that of α-Fe2O3.9,14 In this model, the band gap, which is determined by the difference between the VBM of α-Cr2O3 and conduction band minimum (CBM) of α-Fe2O3, is smaller than that of bulk αFe2O3. Since interaction of the d5 shell of Fe3+ and the d3 shell of Cr3+ depends on both the relative concentrations and the spatial arrangement of Fe and Cr, the electronic structure of (Fe1−xCrx)2O3 solid solutions is more complex than that of the α-Fe2O3/α-Cr2O3 heterostructure. In particular, the electronic structure of an isolated Fe2O3/Cr2O3 interface does not imply any dependence on x. An earlier computational study of ordered bulk FeCrO3 found that even though the VBM is dominated by hybridized O 2p and Cr t2g states and the CBM is dominated by the unoccupied Fe 3d states, which are both lower in energy than the unoccupied Cr 3d states, the band gap of FeCrO3 is close to that of α-Fe2O3.8,10 We note that previous studies considered only the electronic ground state of the system and that band gaps were estimated as differences between one-electron energies for the highest occupied and lowest unoccupied bands. However, this approach is prone to errors because the optical band gaps measured experimentally Received: July 27, 2013 Revised: November 5, 2013 Published: November 8, 2013 25504

dx.doi.org/10.1021/jp407496w | J. Phys. Chem. C 2013, 117, 25504−25512

The Journal of Physical Chemistry C

Article

Figure 1. Calculated spin densities for Fe12−nCrnO18 (n = 0, 1, 6, 10, 11, 12) supercells. The spin density of Cr3+ has a near cubic shape (three unpaired electrons occupying t2g orbitals in the high-spin configuration), and the spin density of Fe3+ has a spherical shape (five unpaired electrons occupying t2g and eg orbitals in the high-spin configuration). Blue (dark) and red (light) colors indicate spin-up and spin-down contributions to spin density, respectively. Lattice parameter c is along the [001] direction.

Table 1. Lattice Constants, Magnetic Moments, and One-Electron Band Gaps (Eg) for α-Fe2O3 and α-Cr2O3 α-Fe2O3

α-Cr2O3

experiment29 GGA+U (U = 5 eV)4 this work (U = 4 eV; J = 1 eV) experiment29 GGA+U (U = 5 eV)4 this work (U = 4 eV, J = 1 eV)

a (Å)

c (Å)

μ (μB/cation)

Eg (eV)

5.035 5.067 5.083 4.951 5.073 5.047

13.747 13.882 13.926 13.566 13.839 13.815

4.9 4.11 4.064 2.76 3.01 2.798

2.0 2.0 1.8 3.4 2.6 3.5

[001] direction in α-Fe2O3 are antiferromagnetically coupled, while the in-plane atoms are ferromagnetically coupled (see Figure 1a). In α-Cr2O3, the atoms along the [001] direction are antiferromagnetically coupled, and the neighboring in-plane atoms are also antiferromagnetically coupled, as shown in Figure 1f. To account for possible magnetic configurations in the mixed system, we first modeled (Fe1−xCrx)2O3 using the periodic model and 30-atom supercells with chemical composition Fe12−nCrnO18 (n = 0, 1, 6, 10, 11, 12). The initial atomic positions for α-Fe2O3 and α-Cr2O3 were taken from experimental data.29 Then, the lattice vectors and atomic positions were fully relaxed to minimize the total energies of the corresponding systems. All periodic model calculations were performed using the Vienna Ab initio Simulation Package (VASP).30 The projected augmented wave (PAW) method was used to approximate the electron-ion potential.31,32 Exchange-correlation effects were treated within the Perdew−Burke−Ernzerhoff (PBE) functional33 form of the GGA. The electronic wave functions were expanded in a plane-wave basis set limited by a cutoff energy of 500 eV. Spin−orbit corrections were not included. Energy minimization and the calculations of the densities of states (DOS) were performed using 7 × 7 × 3 and 11 × 11 × 3 Monkhorst−Pack34 k-point meshes, respectively, with their origins at the Γ point. The energies of self-consistent calculations were converged to 10−6 eV/cell, and the lattice and atomic positions were relaxed until the forces on the ions were less than 0.03 eV/Å. It is well-known that the GGA does not describe the properties of strongly correlated materials very well. To mitigate this problem, we have used the GGA+U approach35 to describe the on-site electronic correlations of the Cr and Fe 3d orbitals, respectively. Parameters U = 4.0 eV and J = 1 eV (see Table 1) gave the best overall agreement with experimentally obtained lattice constants, magnetic moments, and band gaps for both α-Fe2O3 and α-Cr2O3, in agreement with earlier studies.36,37

arise from electronically excited states, in which the binding of the excited electrons and corresponding holes cannot be neglected. For example, the difference between one-electron band gaps and the lowest optical absorption energies in complex oxides, such as LaCrO3, can be as large as 1.5 eV.15 Hence, thus far, there has been no rigorous explanation for the observed band gap reduction in (Fe1−xCrx)2O3 solid solutions. Here, we use density functional theory (DFT) and periodic boundary conditions to determine the atomic structure, density of states, and spin ordering in (Fe1−xCrx)2O3 for several values of x. However, elucidating the crucially important optical properties of these systems requires using methods that go beyond the ground-state electronic structure, i.e., explicit calculations of the electronic excited states.16 To this end, we employ an embedded cluster method,17,18 together with a hybrid exchange-correlation functional, which provide a better description of the electronic structure of wide-band gap semiconductors19,20 than functionals based on the standard Generalized Gradient Approximation (GGA). Similar approaches have been used to study optical absorption in complex oxides,12,15 spectroscopic properties of point defects,21−24 electronic structure of interfaces,25 and excited-staterelated phenomena at surfaces.16,26 Calculations of the optical absorption spectra, and assignment of the physical origin of their characteristic features, are carried out using the timedependent DFT (TDDFT), which has become a very computationally attractive approach for studying excited-state spectra in a wide range of molecular and materials systems.27,28

2. STRUCTURES AND METHODS Both α-Fe2O3 and α-Cr2O3 have corundum-like structures and are antiferromagnetic. However, they have different spin configurations. This results in spin disorder in (Fe1−xCrx)2O3 solid solutions, as illustrated by the calculated spin-density maps shown in Figure 1. The neighboring layers along the 25505



Article

Figure 2. Embedded QM clusters: Fe12−nCrnO45 (n = 0, 1, 6, 10, 11, 12). The small spheres are oxygen atoms, and the large dark and light spheres are Fe and Cr atoms, respectively. The interfacial atoms are not shown.

To calculate the optical absorption spectra for (Fe1−xCrx)2O3 solid solutions, we used an embedded cluster method.17,18,26 Here, the (Fe1−xCrx)2O3 system is represented using a nanocluster composed of ∼75 000 ions, in which the central part is modeled quantum-mechanically (QM cluster), while the remainder of the nanocluster is treated classically. We assume that: (i) the optical transitions take place on a time scale significantly smaller than that associated with atomic displacements, and (ii) the effect of electron polarization outside the QM cluster on the energies of optical transitions inside the QM cluster is negligible. Hence, we adopt a nonpolarizable model for the environment and represent its ions using formal point charges. The nanocluster is constructed in the form of a cuboid of 12 × 12 × 12 supercells obtained using the periodic model described above. Each supercell was complemented with point charges, generated as described in ref 17, to eliminate the lowest electrical moments up to hexadecapole. The electrostatic potential in the central part of the nanocluster, constructed using such supercells, is within 0.01 V of that in the infinite periodic lattice. The QM clusters used in this work are shown in Figure 2. Each Fe12−nCrnO 45 cluster includes all cations of the corresponding Fe12−nCrnO18 (n = 0, 1, 6, 10, 11, 12) supercell and all their nearest-neighboring anions. To prevent artificial polarization of the charge density at the border of the QM clusters, the Fe3+ and Cr3+ ions in a ∼5 Å thick region immediately outside the QM clustersthe interface region are represented using full ion Al3+ effective core pseudopotentials12,38 since Al3+ has the same formal charge and a similar ionic radius to those of Fe3+ and Cr3+. Pople’s 6-31G basis39−41 was used for all atoms. The lowestenergy spin-polarized ground-state configurations were obtained using a fragment approach in combination with the Hartree−Fock (HF) method. The resulting density matrices were used as starting guesses for subsequent calculations of the ground and excited states using the hybrid B3LYP density functional.42,43 Figure 3 shows the dependence of the gap between the highest occupied (HOMO) and lowest unoccupied (LUMO) molecular orbitals in Fe2O3 on the cluster size. We find that the gap is well converged for QM clusters containing 12 Fe atoms, which justifies our choice of the QM clusters (see Figure 2) for the calculations of electronically excited states. We also investigated the dependence of the one-electron band gap in the cluster as a function of the HF exchange contribution in the B3LYP functional (see Figure 5).

Figure 3. One-electron band gaps in α-Fe2O3 calculated using embedded QM clusters containing up to 12 Fe atoms. The fitted value of C is 3.78 eV.

Excited-state calculations were performed on the QM clusters of this size using TDDFT with the hybrid B3LYP density functional as implemented in the NWChem quantum chemistry package.44 Due to the large number of one-electron states forming the valence and the conduction bands of the embedded QM clusters, approximately 5000 excited states contribute to their optical absorption spectra below 5.5 eV. To calculate these spectra, we first used the real-time (RT) TDDFT method,23,45−48 in which the electronic density matrix is propagated in time under a time-dependent Kohn−Sham Hamiltonian. For these calculations, a delta-function electric field was used for each polarization (x, y, z), which simultaneously excites all electronic modes and thus yields the full absorption spectrum. The absorption spectra are computed from the electronic polarizability 2ω S(ω) = Tr[Im α(ω)] 3π where the (diagonal) complex polarizability tensor is computed via Fourier transform of the time-dependent dipole moment μ(t) 1 αii(ω) = μiĩ (ω), i = x , y , z κ Here, κ is the delta-function electric field strength, chosen to be weak enough to ensure linear response. For all simulations, the TD Kohn−Sham equations were integrated using a selfconsistent second-order Magnus propagator, and matrices were exponentiated with a contractive power series. All simulations were run for 12 fs, with a time step of 5 × 10−3 fs, and the dipole moments were damped with an exponential with time constant τ = 4 fs before Fourier transform, resulting in 0.3 eV 25506



Article

these structures have little or no effect on the band gap reduction. To understand the origin of the band gap reduction, we calculated projections of the DOS on atomic orbitals for several Cr/Fe ratios (see Figure 4). First, we examine the electronic

full width at half-maximum (fwhm) Lorentzian broadened spectra. As a time-domain technique, however, RT-TDDFT does not provide insight into the origin of the optical absorption features. To obtain this insight at a reasonable computational cost we separated the occupied states of the valence band into nonoverlapping energy intervals, each 0.13−0.45 eV wide depending on the dominant character of the Kohn−Sham orbitals. Then, all transitions, with excitation energies below 5.5 eV, between each interval and all virtual Kohn−Sham orbitals were calculated using the TDDFT method46,49,50 within the Tamm−Dancoff approximation.51 The superposition of all such calculated excitation energies and the corresponding oscillator strengths define the full excitation spectrum of the system. Finally, the absorption spectra were constructed as a superposition of Lorentzian functions, each representing a single excited state with their position and amplitude determined by the corresponding excitation energy and oscillator strength. The fwhm’s were set to 0.2 eV for all Lorentzians.

3. RESULTS AND DISCUSSION For each (Fe1−xCrx) 2 O3 system, several nonequivalent structural and magnetic configurations of the electronic ground state were considered. The lowest-energy structures and the corresponding spin densities, shown in Figure 1, are consistent with other published results.8,10 If either the Cr or Fe concentration is small, the magnetic configuration is commensurate to that of the pure, end member oxides (see Fe11Cr1O18 and Fe1Cr11O18 configurations in Figures 1b and 1e, respectively). However, once the mole fraction of the minority cation reaches ∼17%, deviations from the ordered magnetic structure appear. Thus, at this and higher second cation concentrations, the electronic structure of the solid solution is determined not only by the spatial distribution of Fe and Cr atoms but also by the spin disorder. The effects of impurity-induced lattice strain and the spin mismatch become apparent in the 50%−50% case (x = 0.5). The most stable atomic arrangement of the Fe6Cr6O18 supercell corresponds to the formation of two phases of bulk-like Fe2O3 and Cr2O3 regions and intermixed Fe−Cr interface layers between them. The spin arrangements in each region are the same as in their respective bulk structures, while the Cr−Fe mixed interface atomic planes adopt the ferromagnetic coupling structure seen in α-Fe2O3 (Figure 1c). This result can be attributed to the strong antiferromagnetic coupling between Fe3+ and Cr3+ in corner-sharing octahedra from neighboring layers.10 The three unpaired electrons in Cr3+ fully fill the t2g orbitals and result in a near cubic shape of the spin density distribution, while the five unpaired electrons in Fe fully fill both the t2g and eg orbitals, resulting in a near spherical spin density distribution. These spin density distributions at the cation sites induce a complex character for the spin polarization at the anion sites which, in turn, contributes to: (i) formation of a complex pattern of metal−metal spin interactions and (ii) a broadening of the O 2p band. In addition, two structures with alternating Fe2O3 and Cr2O3 layers having energies of 5 and 0.1 meV/atom above that of the ground state were found. Therefore, it is likely that these structures coexist at room temperature, effectively resulting in disordered Fe2O3−Cr2O3 solid solutions. According to our calculations, these higher-energy structures have band gaps that are approximately 0.3 and 0.6 eV, respectively, larger than that of the lowest energy 50%−50% configuration. In other words,

Figure 4. Orbital projected densities of states (DOS) of the mixed system with chemical formula Fe12−nCrnO18 (n = 0, 1, 6, 10, 11, 12). The red dashed lines indicate the VBM and CBM of Fe12O18, respectively. The blue dashed line indicates the energy below which the O 2p states become dominant in the DOS of Cr12O18. The black line indicates the low boundary of region III. States above 4 eV are unoccupied. We note that the DOS are not normalized to the total number of electrons.

bands of bulk α-Fe2O3 and α-Cr2O3. As seen in Figure 4a, the occupied Fe t2g and eg states are located deep in the valence band. The VBM and the CBM of α-Fe2O3 are dominated by O 2p states hybridized with occupied Fe 3d eg states and unoccupied Fe 3d t2g* and eg* states, respectively (here and below symbol * indicates unoccupied atomic orbitals). The energy difference between them gives an estimate of the band gap at 1.8 eV. The VBM in α-Cr2O3 is dominated by the Cr t2g states partially hybridized with the O 2p states, while the CBM is dominated by Cr 3d* states. Importantly, the VBM in αCr2O3 is ∼0.1 eV above that of α-Fe2O3, and the band gap of αCr2O3 is 3.5 eV, which is much larger than that of α-Fe2O3. To explore the electronic structure of the top of the valence band further, we considered the Bader atomic charges in the neutral and ionized systems. The differences between these charges for each atom type (see Supporting Information for details) indicate the preferred sites of the hole localizations and are consistent with the character of the VBM as obtained from the analysis of the projected density of states. Thus, the VBM and CBM of the mixed system are determined by the occupied Cr t2g states and unoccupied Fe t2g* states, respectively, as seen in Figure 4b−e. For ease of viewing, we have divided the valence band into three regions as indicated in Figure 4. Region I contains valence states with energies above the VBM of α-Fe2O3; region II is below region I and has a lower energy boundary below which the O 2p states 25507



Article

the VBM approaches that of α-Cr2O3. Although there are occupied states in region I, the significant narrowing of the bottom of the conduction band causes the band gap to increase and finally approach the band gap of bulk α-Cr2O3. The calculated band gap as a function of Cr concentration is consistent with the experimental results.9,52 To gain understanding of the optical absorption properties of solid solutions represented by the Fe12−nCrnO18 supercells, we have used the embedded cluster approach with TDDFT as described above. In these calculations, the HOMO−LUMO gap obtained using the standard B3LYP functional (B3LYP-20) is ∼0.7 eV higher than the experimental value. To improve on this result, we tune the HF exchange contribution in the B3LYP functional. The HOMO−LUMO gaps calculated for the embedded clusters representing α-Fe2O3 (Figure 2a) and αCr2O3 (Figure 2f) using the modified B3LYP functionals with 10% HF (B3LYP-10) are 2.4 and 3.5 eV, respectively, which is very close to the corresponding experimental values. To compensate for the attractive electron−hole response after excitation, we selected the HF contribution so the HOMO− LUMO gap is higher than the experimentally observed optical band gap. To this end, the B3LYP functional with 13% HF contribution (B3LYP-13) was used in all subsequent embedded cluster calculations of the excited state. The DOS calculated for the embedded clusters of the same solid solution configurations as considered above are consistent with the results of the periodic model calculations and, thus, validate the embedded cluster model. Finally, we note that the HOMO−LUMO gaps calculated with all modified B3LYP functionals follow the same trend with Fe/Cr ratio (see Figure 5). The optical absorption spectra were calculated for three embedded clusters corresponding to pure α-Fe2O3, α-Cr2O3, and the 50%−50% Fe−Cr mixed configuration (Figure 2a,c,f). The relative band edge positions are schematically shown in Figure 6. These will assist the analysis of calculated optical absorption spectra shown in Figures 7 and 8. The valence bands are divided into regions I, II, and III, as discussed above. The diagram in Figure 6 is constructed on the basis of the one-electron levels calculated for the embedded QM clusters using the B3LYP density functional. For Fe12O18, O 2p states dominate in both regions II and III. Region I in Cr12O18 and Fe6Cr6O18 is above the VBM of Fe12O18 and is dominated by Cr t2g states. The lowest unoccupied orbitals for Fe12O18 are Fe t2g* and eg* states which are far lower than the virtual Cr 3d states in Cr12O18. For each Fe3+ ion, the five lowest unoccupied orbitals have the same spin and are clearly split into eg and t2g subbands. For each Cr3+ ion in Cr12O18 and Fe6Cr6O18, the ten 3d states are split into three occupied t2g (regions I and II) and two virtual eg* states of the same spin, as well as five virtual t2g* and eg* orbitals of the opposite spin. Virtual Cr 3d states overlap in energy and are thus represented with a single block. Figure 7 shows the absorption spectra for α-Fe2O3, α-Cr2O3, and α-(Fe0.50Cr0.50)2O3 calculated using the RT-TDDFT method. In the case of α-Fe2O3, the computed spectrum captures the absorption features at ∼2.5, ∼3.25, ∼4.0, and ∼5.0 eV to within an error of ∼8%. Similar agreement is seen for the case of α-Cr2O3, including the low-energy feature at ∼2.1 eV as well as the peaks at ∼2.6 and ∼4.5 eV. The features at ∼1.75, ∼2.5, ∼3.2, and ∼4.8 eV in the spectrum of α-(Fe0.50Cr0.50)2O3 are also captured to within ∼5%. The overall shapes of the spectra are also in good agreement with experiment,9,52 revealing a qualitatively correct description of the valence and conduction bands using the embedded cluster method.

become dominant in the DOS of α-Cr2O3; region III includes the states located below region II and above the Fe eg states. Variations of the DOS in region I and at the bottom of the conduction band with Cr concentration affect the band gap, as shown in Figure 5. One can see that in Fe-rich solid solutions

Figure 5. Band gap in (Fe1−xCrx)2O3 as a function of chemical composition: optical absorption measurements52 (solid blue diamond), GGA(PBE)+U periodic model calculations (solid red circles), and B3LYP embedded cluster calculations with 20% (black uptriangles), 13% (solid black down-triangles), and 10% (solid black squares) HF exchange contributions, respectively.

the band gap decreases slightly and reaches its minimum value as the Cr concentration is increased from 0 to 50%. As the Cr content is increased further, the band gap first increases gradually until the Fe concentration becomes small and then jumps sharply to that of α-Cr2O3. To rationalize this trend we examine the effect of the Cr/Fe ratio on the electronic structure of the valence and conduction bands more closely. In Fe11Cr1O18, the Cr-induced perturbation of the top of the valence band and Fe 3d conduction band is negligible, as shown in Figure 4b. We note that the unoccupied Cr 3d states are so high on the energy scale that their effect on the electronic structure of Fe11Cr1O18 is negligible. In the alloys with comparable Cr and Fe concentrations, such as Fe6Cr6O12, the top of the valence band is also dominated by Cr t2g states. However, the VBM has a higher energy than that in pure α-Fe2O3 or α-Cr2O3. We attribute this effect to the magnetic structure of these solid solutions in which the spin configuration is determined by the spatial arrangement of Fe and Cr atoms and is different than that of the bulk α-Fe2O3 and α-Cr2O3. Since the unpaired electrons that are localized predominantly on Fe3+ and Cr3+ ions induce spin polarization of the O2− ions, the spin disorder can be regarded as a perturbation that contributes to a broadening of the O 2p and metal band edges and, therefore, results in a smaller band gap. At the same time, while the smaller Fe content results in a lower magnitude of the DOS at the bottom of the conduction band, the shift of the CBM to higher energies is negligible. Hence, the band gap of the Fe6Cr6O12 system is reduced to 1.5 eV. As the concentration of Cr increases further, for instance in Fe2Cr10O18 and Fe1Cr11O18, the Fe atoms become isolated, and the Fe 3d bands transform into a set of isolated states, which leads to two effects. First, the width of the unoccupied Fe 3d bands decreases sharply, and as a result, the bottom of the conduction band shifts to higher energies. Second, the spins on the Fe3+ ions adopt a configuration like that of bulk α-Cr2O3. Thus, the effects of Fe−Cr hybridization become smaller, and 25508



Article

Figure 8. Optical absorption spectra calculated using TDDFT for the embedded clusters (a) Fe12O45, (b) Fe6Cr6O45, and (c) Cr12O45. The positions and heights of the vertical lines indicate the excitation energies and oscillation strengths, respectively, for dipole allowed transitions. The dark-yellow dashed, black dash-dot, and black dotted lines are the absorption spectra excited within energy regions I, II, and III, respectively. The arrows indicate the types of the transitions, as listed in Table 2. The inset in (c) shows the low-intensity part of the spectrum. See text for details.

Figure 6. Schematics of the optical transition types for (a) Fe12O18, (b) Fe6Cr6O18, and (c) Cr12O18. The dark arrows are transitions to the unoccupied Fe t2g* and eg* states. The light arrows are the transitions to the unoccupied Cr 3d states. The light arrows with dark outlines are transitions from occupied top Cr t2g to unoccupied Fe t2g* and eg* states. The occupied states are separated as indicated by black dashed lines into three regions: I, II, and III. Note that the widths of regions I and II are not to scale.

Table 2. Calculated Peak Positions (eV) and the Character of Optical Transitions from

to

label

Fe12O45

Cr12O45

Fe6Cr6O45

Cr t2g (I, II) O2p (III) O2p (II) O2p (II) O2p (III) O2p (III) Cr t2g (I) Cr t2g (I)

Cr 3d* Cr 3d* Fe t2g* Fe eg* Fe t2g* Fe eg* Fe t2g* Fe eg*

C1 C2 F1 F2 F3 F4 M1 M2

3.0 4.0 4.1 5.0 -

3.5, 3.9, 4.5 >5.0 -

3.5,3.9,4.5 >5.0 3.5 4.2 4.3 5.5 1.9 and 2.6 3.5 and 4.3

consistent with the experimental data.9,52 We note that the positions and relative intensities of peaks F2 and F3 depend on the exact definition of regions II and III in α-Fe2O3. However, the convolution of all four peaks F1, F2, F3, and F4 is insensitive to that choice. In α-Cr2O3, the lowest-energy transitions from occupied Cr 3d states (regions I and II) to Cr 3d* states give a low-intensity feature at ∼2.1 eV (see inset in Figure 8c) and are assigned to the intra-Cr t2g → eg* excitation. It is followed by a broad band with an onset at ∼2.7 eV and a maximum at 5.0 eV, labeled as C1 in Figure 8c. The 2.1 eV feature is sensitive to the choice of the energy interval used in the TDDFT calculations of the excited states (see Section 2). In particular, the interval has to be large enough (∼0.45 eV or greater) to include a representative set of the hybridized Cr 3d and O 2p states at the top of the valence band. Extending the energy interval beyond 0.15 eV does not affect the character of the rest of the optical absorption spectrum and assignment of the bands. The transitions in the 2.7−5.0 eV energy range include spin-allowed Cr 3d → Cr 3d* transitions. As can be seen in Figure 8c, the broad band C1 has three maxima located at 3.5, 3.9, and 4.5 eV; definitive decomposition of this band into sub-bands is complicated by strong intermixing of the virtual eg* and t2g* states. However, based on the similarity of the electronic

Figure 7. Real-time TDDFT optical absorption spectra calculated using embedded clusters (a) Fe12O45, (b) Fe6Cr6O45, and (c) Cr12O45. The inset shows the low-intensity part of the optical absorption spectrum between 1 and 3 eV.

To determine the origins of the various features, we again use the TDDFT method, as described in Section 2. The optical absorption spectra, along with the individual transitions that constitute these spectra, are shown in Figure 8. Transitions from the occupied states with energies below region III to the CBM yield excitation energies as large as 6 eV and will not be discussed here. Table 2 summarizes the characters and peak positions of the various transitions. For α-Fe2O3, there are two types of transitions: O 2p (II,III) → Fe t2g* and O 2p (II,III) → Fe eg*, as shown in Figure 6a. Transitions from O 2p states in region II to Fe t2g* and eg* states form two optical absorption features, shown as the dashdotted curve in Figure 8a, with peak positions at 3.0 and 4.0 eV and labeled as F1 and F2, respectively. Similarly, transitions from O 2p in region III to Fe t2g* and eg* form two features, shown as the dotted curve consisting of peaks F3 and F4 at 4.1 and 5.0 eV, respectively (Figure 8a). The F2 and F3 peaks overlap resulting in a “three-peak” optical absorption spectrum 25509



Article

structures of α-Cr2O3 and LaCrO3,9,15,53 we suggest the following sequence of the absorption bands with increasing excitation energy: intra-Cr t2g → eg*, inter-Cr t2g → t2g*, and inter-Cr t2g → eg*. Earlier experimental studies assigned a broad shoulder at 3.4 eV to the charge transfer O 2p → Cr 3d* transitions.9,53 We find no evidence for this in our calculations. Instead, the O 2p → Cr 3d* transitions appear at the excitation energies above 5 eV; the leading edge of these transitions is labeled as C2. The transitions present in pure α-Fe2O3 and α-Cr2O3 are also present in α-(Fe0.50Cr0.50)2O3. The peak positions and the relative contributions to the absorption spectrum change somewhat relative to the end members. Nevertheless, the characters of these transitions remain similar to those in pure αFe2O3 and α-Cr2O3, and the various features are labeled accordingly in Figure 8b. However, α-(Fe0.50Cr0.50)2O3 shows transitions that are not observed in pure α-Fe2O3 and α-Cr2O3. Specifically, excitations from region I to Fe t2g* and Fe eg* states, labeled as M1 and M2, respectively, are found only in the alloy. Each of these gives rise to two distinct peaks: 1.9 and 2.6 eV for M1 (Cr t2g → Fe t2g*) and 3.5 and 4.3 eV for M2 (Cr t2g → Fe eg*). This splitting is attributed to the effect of the spin structure in α-(Fe0.50Cr0.50)2O3. Indeed, the Cr 3d → Fe 3d* transitions are spin allowed only if the spins of the occupied Cr t2g and Fe t2g states are antiparallel. Due to the electrostatic electron−hole interaction, the smaller the distance between the Cr and Fe atoms, the lower the corresponding excitation energy. Thus, the peaks at 1.9 and 3.5 eV are attributed to the Cr t2g → Fe t2g* and Cr t2g → Fe eg* transitions, respectively, for the nearest antiferromagnetically coupled Cr and Fe atoms. We expect that averaging the Cr → Fe transition energies in a homogeneously mixed (Fe1−xCrx)2O3 solid solution results in no such two-peak character of M1 and M2. However, the values of the excitation energies reported for these peaks here are good indicators of the widths of these Cr → Fe absorption features. Finally, the lowest-energy transition obtained in these calculations is 1.7 eV, which is 0.7 eV lower than the calculated excitation energy of α-Fe2O3 and consistent with experimental observations.9 Our calculated optical absorption spectra are in qualitative agreement with experimental results.9,52 We compared the spectra obtained using the TDDFT (see Figures 7 and 8) and those obtained using the joint density of methods, in which the spectra are calculated as the convolution of the occupied and unoccupied one-electron states weighted with the corresponding transition probabilities in the dipole approximation (see Supporting Information). The overall profiles of the spectra calculated using these two approaches are similar, while the absorption energies obtained in the latter method are a few tenths of an electronvolt larger than those obtained with TDDFT. This difference arises because the joint density of states method does not account for the electrostatic interaction between the excited electron and the corresponding hole and, therefore, provides overestimated optical transition energies. For example, the onset of the C1 band shifts to the higher energy by ∼0.5 eV in pure Cr2O3. This effect is also present in the case of Fe2O3 and Fe-rich (Fe1−xCrx)2O3 solid solutions.

systems are characterized by excitations from occupied Cr t2g to unoccupied Fe t2g* states. We note that Fe2O3 is a chargetransfer insulator, whereas Cr2O3 can be thought of as a Mott− Hubbard insulator in that its band gap is the d−d gap. In a similar vein, the (Fe1−xCrx)2O3 solid solutions can be considered to be Mott−Hubbard insulators, but with the conduction band dominated by Fe 3d orbitals instead of Cr 3d orbitals. Therefore, the magnitude of the gap and the character of the corresponding transitions can vary throughout the material, especially at low Fe concentrations. The Cr t2g states lie above the Fe eg−O 2p hybridized valence state, resulting in a smaller band gap in α(Fe0.50Cr0.50)2O3 than in pure α-Fe2O3. However, this band alignment argument alone cannot explain the observed band gap dependence of (Fe1−xCrx)2O3 on x. Additional factors responsible for band gap reduction are broadening of the band edges induced by Fe/Cr site disorder and the incommensurate spin structures of α-Fe2O3 and α-Cr2O3. We used the TDDFT approach to calculate the optical absorption spectra in pure α-Fe2O3, α-Cr2O3, as well as in α(Fe0.50Cr0.50)2O3 and assigned the character of the main optical absorption bands for photon energies of up to 5.5 eV. These calculations demonstrate that Fe−Cr mixing gives rise to a new spin-allowed optical transition from Cr t2g to Fe t2g* and Fe eg*, which in turn results in a lowering of the optical band gap. Thus, forming solid solutions of materials with offset valence and/or conduction band edges is a plausible approach to tuning the optical gap. Finally, we note that our assignment of the lowenergy part of the Cr2O3 absorption spectrum differs from that in earlier studies. In particular, the broad shoulder at 3.4 eV was assigned to O 2p → Cr 3d* charge-transfer transitions.9,53 In our calculations this part of the spectrum is dominated by Cr t2g → Cr 3d* excitations, while the O 2p → Cr 3d* transitions appear at excitation energies above 5 eV.

■

ASSOCIATED CONTENT

S Supporting Information *

Description of the hole localization and the optical absorption spectra calculated using the joint density of states method. This material is available free of charge via the Internet at http:// pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*Tel.: +44 (0) 20 7679 9936. E-mail: [email protected]. Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Authors thank Sara E. Chamberlin, Tiffany C. Kaspar, Daniel R. Gamelin, and Wei Li Cheah for stimulating discussions. This work was supported by the U.S. Department of Energy, Office of Science, Division of Chemical Sciences, Geosciences, and Biosciences under Award No. 48526, and was performed in the Environmental Molecular Sciences Laboratory, a national scientific user facility sponsored by the U.S. Department of Energy’s Office of Biological and Environmental Research and located at the Pacific Northwest National Laboratory (PNNL).

4. SUMMARY Using periodic and embedded cluster DFT calculations, we have investigated the electronic properties of (Fe1−xCrx)2O3 solid solutions. We have found that the band gaps in the mixed 25510



Article

(21) Mysovsky, A. S.; Sushko, P. V.; Radzhabov, E. A.; Reichling, M.; Shluger, A. L. Structure and Properties of Oxygen Centers in CaF2 Crystals from Ab Initio Embedded Cluster Calculations. Phys. Rev. B 2011, 84, 064133. (22) Pascual, J. L.; Barandiaran, Z.; Seijo, L. Ab Initio Theoretical Study of Luminescence Properties of Pr3+-Doped Lu2O3. Theor. Chem. Acc. 2011, 129, 545−554. (23) Govind, N.; Lopata, K.; Rousseau, R.; Andersen, A.; Kowalski, K. Visible Light Absorption of N-Doped TiO2 Rutile Using (LR/RT)TDDFT and Active Space EOMCCSD Calculations. J. Phys. Chem. Lett. 2011, 2, 2696−2701. (24) Carrasco, J.; Sousa, C.; Illas, F.; Sushko, P. V.; Shluger, A. L. Optical Absorption and Luminescence Energies of F Centers in CaO from Ab Initio Embedded Cluster Calculations. J. Chem. Phys. 2006, 125, 074710. (25) Scanlon, D. O.; Dunnill, C. W.; Buckeridge, J.; Shevlin, S. A.; Logsdail, A. J.; Woodley, S. M.; Catlow, C. R. A.; Powell, M. J.; Palgrave, R. G.; Parkin, I. P.; et al. Band Alignment of Rutile and Anatase TiO2. Nat. Mater. 2013, 12, 798−801. (26) Govind, N.; Sushko, P. V.; Hess, W. P.; Valiev, M.; Kowalski, K. Excitons in Potassium Bromide: A Study Using Embedded TimeDependent Density Functional Theory and Equation-of-Motion Coupled Cluster Methods. Chem. Phys. Lett. 2009, 470, 353−357. (27) Marques, M. A.; Ullrich, C. A.; Nogueira, F.; Rubio, A.; Burke, K.; Gross, E. K. U. E. Time-Dependent Density Functional Theory (Lecture Notes in Physics), 1st ed.; Springer: New York, 2006. (28) Casida, M. E.; Huix-Rotllant, M. Progress in Time-Dependent Density-Functional Theory. Annu. Rev. Phys. Chem. 2012, 63, 287− 323. (29) 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. (30) Kresse, G.; Furthmüller, J. Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set. Phys. Rev. B 1996, 54, 11169. (31) Blöchl, P. E. Projector Augmented-Wave Method. Phys. Rev. B 1994, 50, 17953. (32) Kresse, G.; Joubert, D. From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method. Phys. Rev. B 1999, 59, 1758. (33) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865. (34) Monkhorst, H. J.; Pack, J. D. Special Points for Brillouin-Zone Integrations. Phys. Rev. B 1976, 13, 5188. (35) Anisimov, V. I.; Aryasetiawan, F.; Lichtenstein, A. I. FirstPrinciples Calculations of the Electronic Structure and Spectra of Strongly Correlated Systems: the LDA+U Method. J. Phys: Condens. Matter 1997, 9, 767−808. (36) Rollmann, G.; Rohrbach, A.; Entel, P.; Hafner, J. First-Principles Calculation of the Structure and Magnetic Phases of Hematite. Phys. Rev. B 2004, 69, 165107. (37) Shi, S.; Wysocki, A. L.; Belashchenko, K. D. Magnetism of Chromia from First-Principles Calculations. Phys. Rev. B 2009, 79, 104404. (38) Wadt, W. R.; Hay, P. J. Ab Initio Effective Core Potentials for Molecular Calculations. Potentials for Main Group Elements Na to Bi. J. Chem. Phys. 1985, 82, 284−298. (39) Schuchardt, K. L.; Didier, B. T.; Elsethagen, T.; Sun, L.; Gurumoorthi, V.; Chase, J.; Li, J.; Windus, T. L. Basis Set Exchange: A Community Database for Computational Sciences. J. Chem. Inf. Model. 2007, 47, 1045−1052. (40) Feller, D. The Role of Databases in Support of Computational Chemistry Calculations. J. Comput. Chem. 1996, 17, 1571−1586. (41) EMSL Basis Set Exchange. https://bse.pnl.gov/bse/portal. (42) Lee, C.; Yang, W.; Parr, R. G. Development of the Colle-Salvetti Correlation-Energy Formula into a Functional of the Electron Density. Phys. Rev. B 1988, 37, 785. (43) Becke, A. D. Density-Functional Thermochemistry. III. The Role of Exact Exchange. J. Chem. Phys. 1993, 98, 5648−5652.

PNNL is operated for the Department of Energy by the Battelle Memorial Institute under Contract DE-AC06-76RLO-1830. K.L. acknowledges the William Wiley Postdoctoral Fellowship from EMSL. N.G. acknowledges support from the U.S. Department of Energy, Office of Basic Energy Sciences, under Grant No. DESC0008666 of the SciDAC program. P.V.S. acknowledges support from the Royal Society and EPSRC (Grant EP/H018328/1).

■

REFERENCES

(1) Grätzel, M. Photoelectrochemical Cells. Nature 2001, 414, 338− 344. (2) Pozun, Z. D.; Henkelman, G. Hybrid Density Functional Theory Band Structure Engineering in Hematite. J. Chem. Phys. 2011, 134, 224706. (3) Zaanen, J.; Sawatzky, G.; Allen, J. Band Gaps and Electronic Structure of Transition-Metal Compounds. Phys. Rev. Lett. 1985, 55, 418−421. (4) Rohrbach, A.; Hafner, J.; Kresse, G. Ab Initio Study of the (0001) Surfaces of Hematite and Chromia: Influence of Strong Electronic Correlations. Phys. Rev. B 2004, 70, 125426. (5) Huda, M. N.; Walsh, A.; Yan, Y.; Wei, S.-H.; Al-Jassim, M. M. Electronic, Structural, and Magnetic Effects of 3d Transition Metals in Hematite. J. Appl. Phys. 2010, 107, 123712. (6) Zhao, B.; Kaspar, T. C.; Droubay, T. C.; McCloy, J.; Bowden, M. E.; Shutthanandan, V.; Heald, S. M.; Chambers, S. A. Electrical Transport Properties of Ti-Doped Fe2O3(0001) Epitaxial Films. Phys. Rev. B 2011, 84, 245325. (7) Benny, S.; Grau-Crespo, R.; de Leeuw, N. H. A Theoretical Investigation of α-Fe2O3-Cr2O3 Solid Solutions. Phys. Chem. Chem. Phys. 2009, 11, 808−815. (8) Moore, E. First-Principles Study of the Mixed Oxide α-FeCrO3. Phys. Rev. B 2007, 76, 195107. (9) Mashiko, H.; Oshima, T.; Ohtomo, A. Band-Gap Narrowing in α(CrxFe1−x)2O3 Solid-Solution Films. Appl. Phys. Lett. 2011, 99, 241904. (10) Nabi, H. S.; Pentcheva, R. Energetic Stability and Magnetic Coupling in (Cr1−xFex)2O3: Evidence for a Ferrimagnetic IlmeniteType Superlattice from First Principles. Phys. Rev. B 2011, 83, 214424. (11) Velev, J.; Bandyopadhyay, A.; Butler, W.; Sarker, S. Electronic and Magnetic Structure of Transition-Metal-Doped α-Hematite. Phys. Rev. B 2005, 71, 205208. (12) Liao, P.; Carter, E. A. Optical Excitations in Hematite (α-Fe2O3) via Embedded Cluster Models: A CASPT2 Study. J. Phys. Chem. C 2011, 115, 20795−20805. (13) Stankic, S.; Sterrer, M.; Hofmann, P.; Bernardi, J.; Diwald, O.; Knözinger, E. Novel Optical Surface Properties of Ca2+-Doped MgO Nanocrystals. Nano Lett. 2005, 5, 1889−1893. (14) Chambers, S.; Liang, Y.; Gao, Y. Noncummutative Band Offset at α-Cr2O3/α-Fe2O3(0001) Heterojunctions. Phys. Rev. B 2000, 61, 13223−13229. (15) Sushko, P. V.; Qiao, L.; Bowden, M.; Varga, T.; Exarhos, G. J.; Urban, F. K., III; Barton, D.; Chambers, S. A. Multiband Optical Absorption Controlled by Lattice Strain in Thin-Film LaCrO3. Phys. Rev. Lett. 2013, 110, 077401. (16) Sousa, C.; Tosoni, S.; Illas, F. Theoretical Approaches to Excited-States-Related Phenomena in Oxide Surfaces. Chem. Rev. 2013, 113, 4456−4495. (17) Sushko, P. V.; Abarenkov, I. V. General Purpose Electrostatic Embedding Potential. J. Chem. Theory Comput. 2010, 6, 1323−1333. (18) Muñoz Ramo, D.; Sushko, P. V.; Gavartin, J. L.; Shluger, A. L. Oxygen Vacancies in Cubic ZrO2 Nanocrystals Studied by an Ab Initio Embedded Cluster Method. Phys. Rev. B 2008, 78, 235432. (19) Wang, F.; Di Valentin, C.; Pacchioni, G. Electronic and Structural Properties of WO3: A Systematic Hybrid DFT Study. J. Phys. Chem. C 2011, 115, 8345−8353. (20) Gallino, F.; Di Valentin, C. Copper Impurities in bulk ZnO: a Hybrid Density Functional Study. J. Chem. Phys. 2011, 134, 144506. 25511



Article

(44) Valiev, M.; Bylaska, E. J.; Govind, N.; Kowalski, K.; Straatsma, T. P.; Van Dam, H. J. J.; Wang, D.; Nieplocha, J.; Apra, E.; Windus, T. L.; et al. NWChem: A Comprehensive and Scalable Open-Source Solution for Large Scale Molecular Simulations. Comput. Phys. Commun. 2010, 181, 1477−1489. (45) Sánchez-de-Armas, R.; Oviedo Lόpez, J.; San-Miguel, M. A.; Sanz, J. F.; Ordejόn, P.; Pruneda, M. Real-Time TD-DFT Simulations in Dye Sensitized Solar Cells: The Electronic Absorption Spectrum of Alizarin Supported on TiO2 Nanoclusters. J. Chem. Theory Comput. 2010, 6, 2856−2865. (46) Lopata, K.; Van Kuiken, B. E.; Khalil, M.; Govind, N. LinearResponse and Real-Time Time-Dependent Density Functional Theory Studies of Core-Level Near-Edge X-Ray Absorption. J. Chem. Theory Comput. 2012, 8, 3284−3292. (47) Lopata, K.; Govind, N. Modeling Fast Electron Dynamics with Real-Time Time-Dependent Density Functional Theory: Application to Small Molecules and Chromophores. J. Chem. Theory Comput. 2011, 7, 1344−1355. (48) Li, X.; Tully, J. C.; Schlegel, H. B.; Frisch, M. J. Ab initio Ehrenfest Dynamics. J. Chem. Phys. 2005, 123, 084106. (49) Zhang, Y.; Biggs, J. D.; Healion, D.; Govind, N.; Mukamel, S. Core and Valence Excitations in Resonant X-ray Spectroscopy Using Restricted Excitation Window Time-Dependent Density Functional Theory. J. Chem. Phys. 2012, 137, 194306. (50) Van Kuiken, B. E.; Valiev, M.; Daifuku, S. L.; Bannan, C.; Strader, M. L.; Cho, H.; Huse, N.; Schoenlein, R. W.; Govind, N.; Khalil, M. Simulating Ru L3-Edge X-ray Absorption Spectroscopy with Time-Dependent Density Functional Theory: Model Complexes and Electron Localization in Mixed-Valence Metal Dimers. J. Phys. Chem. A 2013, 117, 4444−4454. (51) Hirata, S.; Head-Gordon, M. Time-Dependent Density Functional Theory Within the Tamm−Dancoff Approximation. Chem. Phys. Lett. 1999, 314, 291−299. (52) Chamberlin, S. E.; Wang, Y.; Lopata, K.; Kaspar, T. C.; Cohn, A. W.; Gamelin, D. R.; Govind, N.; Sushko, P. V.; Chambers, S. A. Optical Absorption and Spectral Photoconductivity in α(Fe1−xCrx)2O3 solid-solution thin films. J. Phys.: Condens. Matter 2013, 25, 392002. (53) Blazey, K. W. The Wavelength-Modulated Reflectivity of Cr2O3 and Its Relation to the Absorption Spectrum of Other Cr3+-Containing Insulators. Solid State Commun. 1972, 11, 371−374.

25512


Optical Absorption and Band Gap Reduction in (Fe1âxCrx) - American

Recommend Documents

Optical Absorption and Band Gap Reduction in (Fe1âxCrx) - American