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Investigation of correlation effects on the electronic structure of 3d1 perovskites
Accepted Manuscript 1 Investigation of correlation effects on the electronic structure of 3d perovskites Spriha Kumari, Aditya K. Sahu, Sanhita Paul, Satyabrata Raj PII:
S0022-3697(18)30921-1
DOI:
10.1016/j.jpcs.2018.09.023
Reference:
PCS 8736
To appear in:
Journal of Physics and Chemistry of Solids
Received Date: 10 April 2018 Revised Date:
15 September 2018
Accepted Date: 17 September 2018
Please cite this article as: S. Kumari, A.K. Sahu, S. Paul, S. Raj, Investigation of correlation effects 1 on the electronic structure of 3d perovskites, Journal of Physics and Chemistry of Solids (2018), doi: https://doi.org/10.1016/j.jpcs.2018.09.023. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Investigation of correlation effects on the electronic structure of 3d1 perovskites Spriha Kumari,1 Aditya K. Sahu,1 Sanhita Paul,1,2 and Satyabrata Raj1,*
Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata,
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1
Mohanpur, Nadia 741246, India 2
Department of Basic Science and Humanities, Institute of Engineering & Management, Sector-
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V, Saltlake, Kolkata 700091, India
Abstract
We have investigated the electronic structure of the 3d1 perovskites SrVO3, CaVO3, and YTiO3 by performing ab initio full-potential linearized augmented plane wave band-structure calculations within the density functional theory framework. The electronic band structure with the correct ground state for 3d1 perovskites could not be realized with the generalized gradient
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approximation (GGA). Although the GGA can explain the metallic ground state of SrVO3 and CaVO3, it fails to find the correct ground state of YTiO3, which is a Mott-Hubbard insulator with a band gap of approximately 0.7eV. Our GGA+U approach with systematic variation of both the electron (U) and the exchange (J) correlation energies has helped in understanding the correct
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electronic structure of these perovskites. We conducted a series of calculations with different Ueff (U − J) in the GGA+U approach to understand the effect of electron correlation on the electronic
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structure. Our procedure ultimately succeeded in describing the insulating electronic structure and correct ferromagnetic ground state of YTiO3.
1. Introduction
Transition metal oxides (TMOs) with a partially filled 3d band behave very differently from usual normal metals because of their narrow 3d band and strong electron-electron correlations, which sometimes leads to an insulating phase known as a Mott insulator [1]. These TMOs exhibit a wide range of interesting electronic and magnetic properties, including interesting phase transitions leading to an insulating state from the metallic state, evolution of
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superconductivity, high resistance with magnetic fields, and also the development of charge density waves as a function of temperature, pressure, doping, or chemical substitution. One of the challenging and fascinating aspects is to understand the electronic structure of such TMOs, which are largely influenced by the strong electron correlation associated with it. Significant
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progress has been made in understanding the electronic structure of such materials with use of various approximations, such as the local density approximation (LDA), the generalized gradient approximation (GGA), and the GGA+U approach, where U is the electron-electron correlation energy. The free-electron model of band theory [2] is not sufficient to describe the ground state
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of many exotic materials because of the strong electron-electron interactions [3]. It has been demonstrated that the strong on-site Coulomb interaction is not well described in LDA-type or
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GGA-type calculations. As a result, the LDA or the GGA does not mostly give the correct ground state in strongly correlated systems. In general, the on-site Coulomb interactions are particularly strong for d and f localized electron systems. The strong interaction of d electrons, also defined as Hubbard U, of transition metal ions should be treated in a proper way to understand the electronic structure of strongly correlated systems. The strength of the on-site interactions is described by two parameters: namely, U (on-site Coulomb interactions) and J (on-
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site exchange interactions). Usually, both U and J are obtained semiempirically by comparison with a physical parameter such as the experimental band gap. Thus, the LDA+U (GGA+U) approach basically accommodates the strong interaction of these d or f electrons in 3d/4f compounds. Several studies with the LDA+U (GGA+U) approach have been performed for
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many transition metal compounds [4–6]. The GGA+U modification can be introduced to the ab initio calculation in two different ways. Either both U and J can be treated independently or Ueff
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= U − J can be considered as a single parameter for Coulomb interaction. Ueff ranges from 4 to 8 eV as reported in several TMOs (i.e., MnO, FeO, CoO, NiO, etc.). We successfully used the GGA+U approach to get the correct insulating ground states for 3d2 systems such as LaVO3 and YVO3. LaVO3 and YVO3 are strongly correlated insulators with Ueff = 3.85 and 3.7 eV, respectively [7]. Our systematic calculations with a variation of correlation strength predicted the correct ground state of LaVO3 and YVO3. The calculated correct ground states in LaVO3 and YVO3 give rise to a band gap of 1.1 and 1.2 eV, respectively. Both LaVO3 and YVO3 are strongly correlated insulating 3d2 systems, whereas the correlation strength (Ueff) slowly increases from SrVO3 to CaVO3 and finally to YTiO3 in our present studies. We have chosen
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systems where the U/W strength increases slowly from a less correlated metal, SrVO3 (U/W < 1), to a highly correlated metal, CaVO3 (U/W ~ 1), and finally to a strongly correlated Mott insulator, YTiO3 (U/W > 1). SrVO3 has an ideal cubic structure with space group
3 , while
CaVO3 and YTiO3 have distorted orthorhombic perovskite structures with space group Pbnm.
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Space groups Pbnm and Pnma are basically similar and can be interchanged by change of the lattice parameters b and c with each other. Because of this interchanging of the space group, one can also find space group Pnma for CaVO3 and YTiO3 in the literature [8]. SrVO3 has an ideal cubic structure with nondistorted corner-sharing octahedra as shown in Fig. 1a, whereas CaVO3
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and YTiO3 exist in the orthorhombic crystal structure. The orthorhombic structure of CaVO3 and YTiO3 can be derived from a super cubic structure (2 × 2 × 2) with lattice parameter a ~ 3.8 Å.
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The orthorhombic structure is due to the rotation, tilting, and deformation of the octahedron from its mean position as shown in Fig. 1b [9, 10]. The V–O–V and Ti–O–Ti bond angles change to 160° and 140° from 180° in CaVO3 and YTiO3, respectively, making YTiO3 the more distorted structure [11, 12]. The octahedral crystal field symmetry splits the degenerate 3d band into t2g and eg orbitals, and the 3d t2g bandwidth, W, is narrowed because of the change in the V–O–V and Ti–O–Ti bond angles in CaVO3 and YTiO3, respectively. As a result, the U/W ratio increases
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in CaVO3 and YTiO3 as compared with SrVO3 because U is expected to be similar for vanadates and titanates [13.] The U/W ratio is defined as the electron correlation strength, and CaVO3 and YTiO3 are more correlated than SrVO3. However, SrVO3 and CaVO3 are still below the critical point for the Mott insulator transition. SrVO3 and CaVO3 belong to the metallic side of the Mott-
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Hubbard transition (U/W < 1) and YTiO3 is on the insulating side of the Mott-Hubbard transition (U/W ~ 1) with a band gap of 0.7 eV. The band gap was determined experimentally from various
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optical studies [14]. In the literature, various theoretical (first-principles calculations using different approximations such as the LDA and the GGA) and experimental (photoemission and inverse photoemission, etc.) studies are available on the electronic structure of SrVO3, CaVO3, and YTiO3 [15–17]. However, theoretical calculations using the LDA and GGA failed to reproduce the experimental results [18, 19]. In SrVO3 and CaVO3 the calculated bandwidth of the V 3d t2g bands is much wider than the experimental value. Splitting of the t2g band into the lower Hubbard band (LHB) and the upper Hubbard band (UHB), which generally happens for strongly correlated systems, is also not reproduced by LDA or GGA calculations. Particularly in YTiO3, the LDA and GGA give similar results and predict it to be a metal, whereas it is well
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known that YTiO3 is a Mott-Hubbard insulator with a band gap of approximately 0.7 eV. LDA+U calculations [20] have significantly reduced the discrepancies between the theoretical and experimental results. The electron (U) and exchange (J) correlation energies of 3d electrons influence dramatically the ground-state properties of strongly correlated systems. In all LDA+U
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(GGA+U) calculations, the electron-electron correlation (U) is an artificial parameter, which is incorporated by hand to match properly the experimental results, but Himmetoglu et al. [20] calculated U (~3.7 eV) using a self-consistent linear response method.
In this work, the electronic structures of SrVO3, CaVO3, and YTiO3 were studied by taking the
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electron-electron correlation effect into account to understand the ground-state properties. To obtain the correct ground state, we adopted the GGA+U formalism and performed calculations
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with U ranging from 1 to 6 eV and with the exchange energy, J, constant. Our motivation was to obtain an experimental quantity such as the band gap correctly from our calculations. We found that our GGA calculations can explain the electronic structure of SrVO3 and CaVO3, whereas GGA+U calculations can well explain the correct magnetic ground state (ferromagnetic insulating) of YTiO3.
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2. Theoretical calculations
We used the cubic crystal structure with space group
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for SrVO3 and the orthorhombic
structure with space group Pbnm for CaVO3 and YTiO3. The lattice parameter a = 3.842 Å for SrVO3 was used in our theoretical calculations [19]. For CaVO3 and YTiO3, we used the lattice
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parameters a = 5.3178 Å, b = 5.3428 Å, and c = 7.5442 Å and a = 5.316 Å, b = 5.679 Å, and c = 7.611 Å, respectively [21, 22]. Ab initio electronic structure calculations by the full-potential linearized augmented plane wave (FP-LAPW) method [23] as described in the Elk code [24]
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were used in our investigation of 3d1 perovskites. Atomic orbitals and plane waves are considered as the basis for the muffin-tin and interstitial regions of the lattice space. In our calculations, we used the GGA (GGA+U) instead of the LDA to understand the electronic structure of the 3d1 system. The LDA considers many-electron systems homogeneously and quite successfully to find the correct ground state of many materials. However, a real system is not always homogeneous, and GGA functionals consider the local density as well as the spatial variation of the density to deal with the inhomogeneity. Generally, GGA calculations improve the total energies, atomization energies, energy barriers, etc. as compared with LDA calculations.
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In many cases, the GGA expands and softens the bonds, and as a result, it sometimes corrects and sometimes overcorrects the LDA results. We used the Perdew-Burke-Ernzerhof GGA functional [25] for our calculations. We also performed band-structure calculations for our 3d1 system using other GGA forms [26, 27] as implemented in the Elk code. We found that all the
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band-structure calculations give quite similar results. We used a 14 × 14 × 14 k-point grid for our GGA and GGA+U calculations. The eigenfunctions, eigenvalues, and total ground-state energy were obtained by our solving self-consistently the Kohn-Sham equations. We constrained the convergence criteria to 1.0 × 10−6 and 1.0 × 10−4 for effective potential and total energy,
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3. Results and discussion 3.1. GGA calculations
To investigate the electronic structure of 3d1 perovskites, initially the GGA of density functional theory was used to calculate the density of states and the electronic band structure near the Fermi energy (EF). We calculated the band structure along all symmetric lines (see the Brillouin zone in Fig. 1c). To obtain the accurate band gap, the total density of states and the partial density of
3.1.1. SrVO3 and CaVO3
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states for each element of the corresponding compound were also calculated.
The valence band structure calculated with the GGA formalism shows that both SrVO3 and
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CaVO3 have bands spread mainly in three bunches as shown in Fig. 2a and b. The number of bands increases in CaVO3 as compared with SrVO3 because a single unit cell of CaVO3 has four
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formula units (Z = 4), whereas SrVO3 has only one formula unit (Z = 1). The valence band of both SrVO3 and CaVO3 is dominated by O 2p character (binding energy between 2 and 7 eV), whereas the V 3d orbitals contribute to the conduction band. A series of photoemission experiments on SrVO3 and CaVO3 [16, 28, 29] suggest that the experimental density of states around EF (up to a binding energy of approximately 1.5 eV) is mainly contributed by a V 3d– derived spectral function. Our GGA calculations also predict the same. The calculated band (binding energy of 2–8 eV) is derived mainly from O 2p character and grossly matches the observed experimental photoemission spectra. Crystal field symmetry removes the V 3d band degeneracy and splits the band into low-energy triply degenerate t2g orbitals and high-energy
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doubly degenerate eg orbitals. Degeneracy of the bands is further lifted in CaVO3 because of the orthorhombic crystal structure, whereas this is not the case for SrVO3 because of its perfect cubic structure. It has been observed that EF lies in the conduction t2g band, which means that both SrVO3 and CaVO3 are metallic as per the GGA calculations. Further, there is no splitting of the
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t2g band into the LHB and the UHB because of strong electron interaction as the GGA formalism underestimates the electron-electron correlations in strongly correlated systems. From our bandstructure calculations, we found that the calculated bandwidth of the t2g orbital is around 2.2 eV for CaVO3 and 2.6 eV for SrVO3, values that are in good agreement with previous LDA
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calculations [14].
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3.1.2. YTiO3
To understand the electronic structure of YTiO3, we performed GGA calculations by the FPLAPW method. The band structure along the high-symmetry direction of the Brillouin zone and the partial density of states for Ti 3d and O 2p contributions are shown in Fig. 2c. The density of states of YTiO3 clearly shows that the Ti 3d band splits into t2g and eg bands because of octahedral crystal field symmetry and EF lies within the t2g band, implying metallic character of
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YTiO3. In reality, YTiO3 is an insulating system, and in our calculations the GGA failed to find the correct ground-state electronic structure. In YTiO3, the valence band is mostly due to the O 2p band, whereas the conduction band is dominated by the Ti 3d t2g band, and the eg band is above the t2g band as in other 3d1 compounds described before. The gross band structures of
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orthorhombic CaVO3 and orthorhombic YTiO3 are similar in nature. The p-d charge transfer energy (∆ = εd − εp) is larger in YTiO3 than in CaVO3, although the absolute energy gap is very
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much underestimated in density functional theory calculations.
3.2. GGA+U calculations with variation of U and J All our 3d1 perovskites are in the vicinity of the Mott-Hubbard transition, and it is known that the LDA and GGA fail to predict the correct electronic structure in systems with strong electronelectron interactions. Therefore, to see the effect of strong electron interactions on the electronic structure of these 3d1 perovskites, we performed GGA+U calculations to capture the correct ground state. Many GGA+U calculations were performed by our keeping J fixed and varying U by assuming that J does not influence the electronic structure as much as U.
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3.2.1. SrVO3 and CaVO3 We performed several GGA+U calculations of SrVO3 and CaVO3 by changing Ueff (= U − J ) from 0 to 5 eV to understand the effect of electron-electron correlation. In these calculations, we
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fixed the exchange-correlation energy, J, to 1 eV. We found from our calculations that both compounds remain in the metallic state even for Ueff = 5 eV, which means they do not show a metal-to-insulator transition. The electronic structure (both the band structure and the density of states) of SrVO3 do not show much change with U. Since SrVO3 has a perfect cubic crystal
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structure with a V–O–V bond angle of 180° (making U/W < 1), there is no splitting of its t2g orbitals into the LHB and UHB (i.e., they are degenerate). However, we observed in our GGA+U
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calculations that the density of states of the t2g orbital is depleted at the Fermi energy for the highest value of the electron-electron correlation strength (U = 5 eV) in CaVO3. This implies that band narrowing (~15%) induced by the decrease of the V–O–V bond angle in CaVO3 (~160°) enhances the electron correlation, hence the dip in the density of states (not shown) of the t2g orbital but not to the extent of metal insulator transition. Since SrVO3 and CaVO3 are metallic, our GGA+U method can well explain the electronic structure. We observed a dip in the density
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of states of the t2g orbital in CaVO3 for the highest value of electron-electron interactions in our GGA+U calculations. This indicates that CaVO3 is a more strongly correlated metallic compound as compared with SrVO3. This prediction is also observed from experimental electronic specific-heat measurements. The specific heat is approximately 15% smaller in SrVO3
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SrVO3. [30]
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than in CaVO3 because of the approximately 15% increase in bandwidth (W) in the case of
3.2.2. YTiO3
We considered YTiO3 to understand the strong correlation effect on a 3d1 system. It has been reported that the Ti–O–Ti angle is 140°, and as a result, it reduces the bandwidth by approximately 20% as compared with SrVO3 [31]. Because of this strong correlation effect YTiO3 is an insulator, unlike SrVO3 and CaVO3 [32-36]. Our previous GGA calculations predicted YTiO3 to be metallic, which is not the correct ground state. To obtain the correct ground state, we performed a systematic procedure under the GGA+U formalism to find the appropriate electron (U) and exchange (J) correlation energies. Sawada and Terakura [34] and
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Okatov et al. [36] considered Ueff = 3–4 eV to calculate the ground state of YTiO3. We found that when we consider electron correlation in GGA+U calculations the band gap opens up at a higher value of Ueff, and we plot the band gap versus Ueff in Fig. 3a. Initially, calculations were performed by our keeping J constant (1 eV) and varying U. Ueff = 0 eV corresponds to the GGA
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calculation that gives the incorrect ground state of YTiO3 as discussed previously. YTiO3 remains metallic until Ueff = 2 eV, and band gap opens up on further increase of Ueff. This indicates that YTiO3 is more like a Mott-Hubbard insulator. For Ueff = 3–4 eV, we found that the band gap increases from 0.12 to 0.48 eV, which is well below the experimental band gap of approximately
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0.7 eV. Further increase of Ueff to 5 eV gives an effectively higher band gap (~0.89 eV) than the experimental value. So the optimized Ueff lies between 4 and 5 eV, and detailed smaller-step
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calculations (Ueff = 4.5–5 eV) were performed (see the inset in Fig. 3a). The inset shows that for YTiO3, Ueff = 4.55 eV is the best parameter to give the correct band gap of approximately 0.7 eV. To see how the band gap of YTiO3 changes with J, we fine-tuned J in a smaller step of 0.1 eV, keeping the total Ueff constant at 4.55 eV as shown in Fig. 3b. We found that for J = 0.1–0.5 eV, the theoretical band gap was 0.65 eV which is lower than the reported value of the band gap. On the other hand, for J = 0.6–1 eV, it closely matches 0.7 eV, and hence we decided to use J =
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0.6 eV and Ueff = 4.55 eV (U = 5.15 eV and J = 0.6 eV) as our optimized values. The FP-LAPW band structure with the parameters U = 5.15 eV and J = 0.6 eV for YTiO3 is shown in Fig. 4. The band structure clearly shows the opening of a gap at EF, indicating the insulating ground state of YTiO3. Because of the strong electron-electron interactions, the t2g band splits into the LHB and
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the UHB, and our GGA+U calculations correctly capture the opening of the band gap across EF.
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3.3. Magnetic ordering in YTiO3
It is worth exploring the spin ordering of YTiO3 by use of our optimized band-structure parameters. We used the optimized Ueff in the GGA+U approach to determine the magnetic ground state of YTiO3. The ground-state energy for paramagnetic, ferromagnetic, and various types (C, A, and G) of antiferromagnetic ordering configurations were calculated as shown in Fig. 5. In perovskite-type oxides, antiferromagnetic ordering can be of three types but energetically they are very close to each other. From the literature [9] it has been found that the ground state of YTiO3 is ferromagnetic with a Curie temperature of 30 K. From our calculations it was found that all magnetic calculations give values that are energetically very close to each
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other, but nevertheless, our magnetic calculations for YTiO3 show that it has the lowest energy in the ferromagnetic spin-ordered state. Hence, our GGA+U calculations can predict the ferromagnetic ground state of YTiO3 as reported in the literature.
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4. Conclusion
In conclusion, we studied the electronic structure of the 3d1 perovskites SrVO3, CaVO3, and YTiO3 by performing self-consistent ab initio calculations using the FP-LAPW method. We used both the GGA and the GGA+U approach to establish the correct ground state of these systems.
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We found that although the GGA can explain the metallic ground state of SrVO3 and CaVO3, it cannot predict that YTiO3 should be an insulator. Our approach of changing U and J
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systematically establishes the correct correlation strength, Ueff, in YTiO3 and describes the correct insulating ferromagnetic ground state with a band gap of approximately 0.7 eV. Our systematic procedure to find the appropriate electron (U) and exchange (J) correlation not only predicts the correct band gap and ground-state magnetic structure but also correctly calculates the energy differences of different magnetic structures, which were found to be quite small
Acknowledgment
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owing to the low Curie temperature of approximately 30 K.
SK (fellow code no. IF140119) acknowledges the Inspire fellowship program of the Department
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Fig. 1. (Color online) (a) Super cell 2 × 2× 2 ABO3 perovskite structure. The B atom is shared by six oxygen atoms forming the octahedron. Because of the deformation of the octahedron, orthorhombic CaVO3 and YTiO3 structures (red boundary) are formed, which are derived from
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the super cell. (b) Orthorhombic ABO3 structure derived from the 2 × 2 × 2 super cell. (c) Cubic and orthorhombic Brillouin zone showing highly symmetric lines. Fig. 2. (Color online) Band structure of (a) cubic SrVO3, (b) orthorhombic CaVO3, and (c) orthorhombic YTiO3 obtained with the generalized gradient approximation. The density of states
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(DOS) and its decomposition into the partial DOS (PDOS) are also shown. Fig. 3.(Color online) (a) Variation of band gap with Ueff for all the 3d1 perovskites. The inset
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shows the narrow region of Ueff for YTiO3. (b) Variation of exchange-correlation energy (J) from 0.1 to 1 eV with the band gap. Fig. 4. (Color online) Electronic structure of orthorhombic YTiO3 calculated by the generalized gradient approximation plus U approach. The density of states (DOS) and its decomposition into the partial DOS (PDOS) are also shown. LHB, lower Hubbard band; UHB, upper Hubbard band. Fig. 5. Total ground-state energies calculated by the generalized gradient approximation plus U approach per unit formula of YTiO3 for ferromagnetic (FM) C-, A-, and G-type antiferromagnetic (AFM) ordering, and paramagnetic (PM) configurations.
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Highlights •
Electronic structure calculations of SrVO3, CaVO3, and YTiO3 by the full-potential linearized augmented plane wave method. Correct ground state with generalized gradient approximation plus U calculations.
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Optimal on-site Coulomb potential predicts the correct band gap and magnetic ground
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state of YTiO3.
S0022-3697(18)30921-1
DOI:
10.1016/j.jpcs.2018.09.023
Reference:
PCS 8736
To appear in:
Journal of Physics and Chemistry of Solids
Received Date: 10 April 2018 Revised Date:
15 September 2018
Accepted Date: 17 September 2018
Please cite this article as: S. Kumari, A.K. Sahu, S. Paul, S. Raj, Investigation of correlation effects 1 on the electronic structure of 3d perovskites, Journal of Physics and Chemistry of Solids (2018), doi: https://doi.org/10.1016/j.jpcs.2018.09.023. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Investigation of correlation effects on the electronic structure of 3d1 perovskites Spriha Kumari,1 Aditya K. Sahu,1 Sanhita Paul,1,2 and Satyabrata Raj1,*
Department of Physical Sciences, Indian Institute of Science Education and Research Kolkata,
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1
Mohanpur, Nadia 741246, India 2
Department of Basic Science and Humanities, Institute of Engineering & Management, Sector-
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V, Saltlake, Kolkata 700091, India
Abstract
We have investigated the electronic structure of the 3d1 perovskites SrVO3, CaVO3, and YTiO3 by performing ab initio full-potential linearized augmented plane wave band-structure calculations within the density functional theory framework. The electronic band structure with the correct ground state for 3d1 perovskites could not be realized with the generalized gradient
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approximation (GGA). Although the GGA can explain the metallic ground state of SrVO3 and CaVO3, it fails to find the correct ground state of YTiO3, which is a Mott-Hubbard insulator with a band gap of approximately 0.7eV. Our GGA+U approach with systematic variation of both the electron (U) and the exchange (J) correlation energies has helped in understanding the correct
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electronic structure of these perovskites. We conducted a series of calculations with different Ueff (U − J) in the GGA+U approach to understand the effect of electron correlation on the electronic
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structure. Our procedure ultimately succeeded in describing the insulating electronic structure and correct ferromagnetic ground state of YTiO3.
1. Introduction
Transition metal oxides (TMOs) with a partially filled 3d band behave very differently from usual normal metals because of their narrow 3d band and strong electron-electron correlations, which sometimes leads to an insulating phase known as a Mott insulator [1]. These TMOs exhibit a wide range of interesting electronic and magnetic properties, including interesting phase transitions leading to an insulating state from the metallic state, evolution of
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superconductivity, high resistance with magnetic fields, and also the development of charge density waves as a function of temperature, pressure, doping, or chemical substitution. One of the challenging and fascinating aspects is to understand the electronic structure of such TMOs, which are largely influenced by the strong electron correlation associated with it. Significant
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progress has been made in understanding the electronic structure of such materials with use of various approximations, such as the local density approximation (LDA), the generalized gradient approximation (GGA), and the GGA+U approach, where U is the electron-electron correlation energy. The free-electron model of band theory [2] is not sufficient to describe the ground state
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of many exotic materials because of the strong electron-electron interactions [3]. It has been demonstrated that the strong on-site Coulomb interaction is not well described in LDA-type or
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GGA-type calculations. As a result, the LDA or the GGA does not mostly give the correct ground state in strongly correlated systems. In general, the on-site Coulomb interactions are particularly strong for d and f localized electron systems. The strong interaction of d electrons, also defined as Hubbard U, of transition metal ions should be treated in a proper way to understand the electronic structure of strongly correlated systems. The strength of the on-site interactions is described by two parameters: namely, U (on-site Coulomb interactions) and J (on-
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site exchange interactions). Usually, both U and J are obtained semiempirically by comparison with a physical parameter such as the experimental band gap. Thus, the LDA+U (GGA+U) approach basically accommodates the strong interaction of these d or f electrons in 3d/4f compounds. Several studies with the LDA+U (GGA+U) approach have been performed for
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many transition metal compounds [4–6]. The GGA+U modification can be introduced to the ab initio calculation in two different ways. Either both U and J can be treated independently or Ueff
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= U − J can be considered as a single parameter for Coulomb interaction. Ueff ranges from 4 to 8 eV as reported in several TMOs (i.e., MnO, FeO, CoO, NiO, etc.). We successfully used the GGA+U approach to get the correct insulating ground states for 3d2 systems such as LaVO3 and YVO3. LaVO3 and YVO3 are strongly correlated insulators with Ueff = 3.85 and 3.7 eV, respectively [7]. Our systematic calculations with a variation of correlation strength predicted the correct ground state of LaVO3 and YVO3. The calculated correct ground states in LaVO3 and YVO3 give rise to a band gap of 1.1 and 1.2 eV, respectively. Both LaVO3 and YVO3 are strongly correlated insulating 3d2 systems, whereas the correlation strength (Ueff) slowly increases from SrVO3 to CaVO3 and finally to YTiO3 in our present studies. We have chosen
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systems where the U/W strength increases slowly from a less correlated metal, SrVO3 (U/W < 1), to a highly correlated metal, CaVO3 (U/W ~ 1), and finally to a strongly correlated Mott insulator, YTiO3 (U/W > 1). SrVO3 has an ideal cubic structure with space group
3 , while
CaVO3 and YTiO3 have distorted orthorhombic perovskite structures with space group Pbnm.
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Space groups Pbnm and Pnma are basically similar and can be interchanged by change of the lattice parameters b and c with each other. Because of this interchanging of the space group, one can also find space group Pnma for CaVO3 and YTiO3 in the literature [8]. SrVO3 has an ideal cubic structure with nondistorted corner-sharing octahedra as shown in Fig. 1a, whereas CaVO3
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and YTiO3 exist in the orthorhombic crystal structure. The orthorhombic structure of CaVO3 and YTiO3 can be derived from a super cubic structure (2 × 2 × 2) with lattice parameter a ~ 3.8 Å.
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The orthorhombic structure is due to the rotation, tilting, and deformation of the octahedron from its mean position as shown in Fig. 1b [9, 10]. The V–O–V and Ti–O–Ti bond angles change to 160° and 140° from 180° in CaVO3 and YTiO3, respectively, making YTiO3 the more distorted structure [11, 12]. The octahedral crystal field symmetry splits the degenerate 3d band into t2g and eg orbitals, and the 3d t2g bandwidth, W, is narrowed because of the change in the V–O–V and Ti–O–Ti bond angles in CaVO3 and YTiO3, respectively. As a result, the U/W ratio increases
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in CaVO3 and YTiO3 as compared with SrVO3 because U is expected to be similar for vanadates and titanates [13.] The U/W ratio is defined as the electron correlation strength, and CaVO3 and YTiO3 are more correlated than SrVO3. However, SrVO3 and CaVO3 are still below the critical point for the Mott insulator transition. SrVO3 and CaVO3 belong to the metallic side of the Mott-
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Hubbard transition (U/W < 1) and YTiO3 is on the insulating side of the Mott-Hubbard transition (U/W ~ 1) with a band gap of 0.7 eV. The band gap was determined experimentally from various
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optical studies [14]. In the literature, various theoretical (first-principles calculations using different approximations such as the LDA and the GGA) and experimental (photoemission and inverse photoemission, etc.) studies are available on the electronic structure of SrVO3, CaVO3, and YTiO3 [15–17]. However, theoretical calculations using the LDA and GGA failed to reproduce the experimental results [18, 19]. In SrVO3 and CaVO3 the calculated bandwidth of the V 3d t2g bands is much wider than the experimental value. Splitting of the t2g band into the lower Hubbard band (LHB) and the upper Hubbard band (UHB), which generally happens for strongly correlated systems, is also not reproduced by LDA or GGA calculations. Particularly in YTiO3, the LDA and GGA give similar results and predict it to be a metal, whereas it is well
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known that YTiO3 is a Mott-Hubbard insulator with a band gap of approximately 0.7 eV. LDA+U calculations [20] have significantly reduced the discrepancies between the theoretical and experimental results. The electron (U) and exchange (J) correlation energies of 3d electrons influence dramatically the ground-state properties of strongly correlated systems. In all LDA+U
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(GGA+U) calculations, the electron-electron correlation (U) is an artificial parameter, which is incorporated by hand to match properly the experimental results, but Himmetoglu et al. [20] calculated U (~3.7 eV) using a self-consistent linear response method.
In this work, the electronic structures of SrVO3, CaVO3, and YTiO3 were studied by taking the
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electron-electron correlation effect into account to understand the ground-state properties. To obtain the correct ground state, we adopted the GGA+U formalism and performed calculations
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with U ranging from 1 to 6 eV and with the exchange energy, J, constant. Our motivation was to obtain an experimental quantity such as the band gap correctly from our calculations. We found that our GGA calculations can explain the electronic structure of SrVO3 and CaVO3, whereas GGA+U calculations can well explain the correct magnetic ground state (ferromagnetic insulating) of YTiO3.
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2. Theoretical calculations
We used the cubic crystal structure with space group
3
for SrVO3 and the orthorhombic
structure with space group Pbnm for CaVO3 and YTiO3. The lattice parameter a = 3.842 Å for SrVO3 was used in our theoretical calculations [19]. For CaVO3 and YTiO3, we used the lattice
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parameters a = 5.3178 Å, b = 5.3428 Å, and c = 7.5442 Å and a = 5.316 Å, b = 5.679 Å, and c = 7.611 Å, respectively [21, 22]. Ab initio electronic structure calculations by the full-potential linearized augmented plane wave (FP-LAPW) method [23] as described in the Elk code [24]
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were used in our investigation of 3d1 perovskites. Atomic orbitals and plane waves are considered as the basis for the muffin-tin and interstitial regions of the lattice space. In our calculations, we used the GGA (GGA+U) instead of the LDA to understand the electronic structure of the 3d1 system. The LDA considers many-electron systems homogeneously and quite successfully to find the correct ground state of many materials. However, a real system is not always homogeneous, and GGA functionals consider the local density as well as the spatial variation of the density to deal with the inhomogeneity. Generally, GGA calculations improve the total energies, atomization energies, energy barriers, etc. as compared with LDA calculations.
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In many cases, the GGA expands and softens the bonds, and as a result, it sometimes corrects and sometimes overcorrects the LDA results. We used the Perdew-Burke-Ernzerhof GGA functional [25] for our calculations. We also performed band-structure calculations for our 3d1 system using other GGA forms [26, 27] as implemented in the Elk code. We found that all the
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band-structure calculations give quite similar results. We used a 14 × 14 × 14 k-point grid for our GGA and GGA+U calculations. The eigenfunctions, eigenvalues, and total ground-state energy were obtained by our solving self-consistently the Kohn-Sham equations. We constrained the convergence criteria to 1.0 × 10−6 and 1.0 × 10−4 for effective potential and total energy,
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3. Results and discussion 3.1. GGA calculations
To investigate the electronic structure of 3d1 perovskites, initially the GGA of density functional theory was used to calculate the density of states and the electronic band structure near the Fermi energy (EF). We calculated the band structure along all symmetric lines (see the Brillouin zone in Fig. 1c). To obtain the accurate band gap, the total density of states and the partial density of
3.1.1. SrVO3 and CaVO3
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states for each element of the corresponding compound were also calculated.
The valence band structure calculated with the GGA formalism shows that both SrVO3 and
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CaVO3 have bands spread mainly in three bunches as shown in Fig. 2a and b. The number of bands increases in CaVO3 as compared with SrVO3 because a single unit cell of CaVO3 has four
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formula units (Z = 4), whereas SrVO3 has only one formula unit (Z = 1). The valence band of both SrVO3 and CaVO3 is dominated by O 2p character (binding energy between 2 and 7 eV), whereas the V 3d orbitals contribute to the conduction band. A series of photoemission experiments on SrVO3 and CaVO3 [16, 28, 29] suggest that the experimental density of states around EF (up to a binding energy of approximately 1.5 eV) is mainly contributed by a V 3d– derived spectral function. Our GGA calculations also predict the same. The calculated band (binding energy of 2–8 eV) is derived mainly from O 2p character and grossly matches the observed experimental photoemission spectra. Crystal field symmetry removes the V 3d band degeneracy and splits the band into low-energy triply degenerate t2g orbitals and high-energy
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doubly degenerate eg orbitals. Degeneracy of the bands is further lifted in CaVO3 because of the orthorhombic crystal structure, whereas this is not the case for SrVO3 because of its perfect cubic structure. It has been observed that EF lies in the conduction t2g band, which means that both SrVO3 and CaVO3 are metallic as per the GGA calculations. Further, there is no splitting of the
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t2g band into the LHB and the UHB because of strong electron interaction as the GGA formalism underestimates the electron-electron correlations in strongly correlated systems. From our bandstructure calculations, we found that the calculated bandwidth of the t2g orbital is around 2.2 eV for CaVO3 and 2.6 eV for SrVO3, values that are in good agreement with previous LDA
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calculations [14].
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3.1.2. YTiO3
To understand the electronic structure of YTiO3, we performed GGA calculations by the FPLAPW method. The band structure along the high-symmetry direction of the Brillouin zone and the partial density of states for Ti 3d and O 2p contributions are shown in Fig. 2c. The density of states of YTiO3 clearly shows that the Ti 3d band splits into t2g and eg bands because of octahedral crystal field symmetry and EF lies within the t2g band, implying metallic character of
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YTiO3. In reality, YTiO3 is an insulating system, and in our calculations the GGA failed to find the correct ground-state electronic structure. In YTiO3, the valence band is mostly due to the O 2p band, whereas the conduction band is dominated by the Ti 3d t2g band, and the eg band is above the t2g band as in other 3d1 compounds described before. The gross band structures of
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orthorhombic CaVO3 and orthorhombic YTiO3 are similar in nature. The p-d charge transfer energy (∆ = εd − εp) is larger in YTiO3 than in CaVO3, although the absolute energy gap is very
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much underestimated in density functional theory calculations.
3.2. GGA+U calculations with variation of U and J All our 3d1 perovskites are in the vicinity of the Mott-Hubbard transition, and it is known that the LDA and GGA fail to predict the correct electronic structure in systems with strong electronelectron interactions. Therefore, to see the effect of strong electron interactions on the electronic structure of these 3d1 perovskites, we performed GGA+U calculations to capture the correct ground state. Many GGA+U calculations were performed by our keeping J fixed and varying U by assuming that J does not influence the electronic structure as much as U.
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3.2.1. SrVO3 and CaVO3 We performed several GGA+U calculations of SrVO3 and CaVO3 by changing Ueff (= U − J ) from 0 to 5 eV to understand the effect of electron-electron correlation. In these calculations, we
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fixed the exchange-correlation energy, J, to 1 eV. We found from our calculations that both compounds remain in the metallic state even for Ueff = 5 eV, which means they do not show a metal-to-insulator transition. The electronic structure (both the band structure and the density of states) of SrVO3 do not show much change with U. Since SrVO3 has a perfect cubic crystal
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structure with a V–O–V bond angle of 180° (making U/W < 1), there is no splitting of its t2g orbitals into the LHB and UHB (i.e., they are degenerate). However, we observed in our GGA+U
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calculations that the density of states of the t2g orbital is depleted at the Fermi energy for the highest value of the electron-electron correlation strength (U = 5 eV) in CaVO3. This implies that band narrowing (~15%) induced by the decrease of the V–O–V bond angle in CaVO3 (~160°) enhances the electron correlation, hence the dip in the density of states (not shown) of the t2g orbital but not to the extent of metal insulator transition. Since SrVO3 and CaVO3 are metallic, our GGA+U method can well explain the electronic structure. We observed a dip in the density
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of states of the t2g orbital in CaVO3 for the highest value of electron-electron interactions in our GGA+U calculations. This indicates that CaVO3 is a more strongly correlated metallic compound as compared with SrVO3. This prediction is also observed from experimental electronic specific-heat measurements. The specific heat is approximately 15% smaller in SrVO3
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SrVO3. [30]
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than in CaVO3 because of the approximately 15% increase in bandwidth (W) in the case of
3.2.2. YTiO3
We considered YTiO3 to understand the strong correlation effect on a 3d1 system. It has been reported that the Ti–O–Ti angle is 140°, and as a result, it reduces the bandwidth by approximately 20% as compared with SrVO3 [31]. Because of this strong correlation effect YTiO3 is an insulator, unlike SrVO3 and CaVO3 [32-36]. Our previous GGA calculations predicted YTiO3 to be metallic, which is not the correct ground state. To obtain the correct ground state, we performed a systematic procedure under the GGA+U formalism to find the appropriate electron (U) and exchange (J) correlation energies. Sawada and Terakura [34] and
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Okatov et al. [36] considered Ueff = 3–4 eV to calculate the ground state of YTiO3. We found that when we consider electron correlation in GGA+U calculations the band gap opens up at a higher value of Ueff, and we plot the band gap versus Ueff in Fig. 3a. Initially, calculations were performed by our keeping J constant (1 eV) and varying U. Ueff = 0 eV corresponds to the GGA
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calculation that gives the incorrect ground state of YTiO3 as discussed previously. YTiO3 remains metallic until Ueff = 2 eV, and band gap opens up on further increase of Ueff. This indicates that YTiO3 is more like a Mott-Hubbard insulator. For Ueff = 3–4 eV, we found that the band gap increases from 0.12 to 0.48 eV, which is well below the experimental band gap of approximately
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0.7 eV. Further increase of Ueff to 5 eV gives an effectively higher band gap (~0.89 eV) than the experimental value. So the optimized Ueff lies between 4 and 5 eV, and detailed smaller-step
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calculations (Ueff = 4.5–5 eV) were performed (see the inset in Fig. 3a). The inset shows that for YTiO3, Ueff = 4.55 eV is the best parameter to give the correct band gap of approximately 0.7 eV. To see how the band gap of YTiO3 changes with J, we fine-tuned J in a smaller step of 0.1 eV, keeping the total Ueff constant at 4.55 eV as shown in Fig. 3b. We found that for J = 0.1–0.5 eV, the theoretical band gap was 0.65 eV which is lower than the reported value of the band gap. On the other hand, for J = 0.6–1 eV, it closely matches 0.7 eV, and hence we decided to use J =
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0.6 eV and Ueff = 4.55 eV (U = 5.15 eV and J = 0.6 eV) as our optimized values. The FP-LAPW band structure with the parameters U = 5.15 eV and J = 0.6 eV for YTiO3 is shown in Fig. 4. The band structure clearly shows the opening of a gap at EF, indicating the insulating ground state of YTiO3. Because of the strong electron-electron interactions, the t2g band splits into the LHB and
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the UHB, and our GGA+U calculations correctly capture the opening of the band gap across EF.
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3.3. Magnetic ordering in YTiO3
It is worth exploring the spin ordering of YTiO3 by use of our optimized band-structure parameters. We used the optimized Ueff in the GGA+U approach to determine the magnetic ground state of YTiO3. The ground-state energy for paramagnetic, ferromagnetic, and various types (C, A, and G) of antiferromagnetic ordering configurations were calculated as shown in Fig. 5. In perovskite-type oxides, antiferromagnetic ordering can be of three types but energetically they are very close to each other. From the literature [9] it has been found that the ground state of YTiO3 is ferromagnetic with a Curie temperature of 30 K. From our calculations it was found that all magnetic calculations give values that are energetically very close to each
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other, but nevertheless, our magnetic calculations for YTiO3 show that it has the lowest energy in the ferromagnetic spin-ordered state. Hence, our GGA+U calculations can predict the ferromagnetic ground state of YTiO3 as reported in the literature.
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4. Conclusion
In conclusion, we studied the electronic structure of the 3d1 perovskites SrVO3, CaVO3, and YTiO3 by performing self-consistent ab initio calculations using the FP-LAPW method. We used both the GGA and the GGA+U approach to establish the correct ground state of these systems.
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We found that although the GGA can explain the metallic ground state of SrVO3 and CaVO3, it cannot predict that YTiO3 should be an insulator. Our approach of changing U and J
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systematically establishes the correct correlation strength, Ueff, in YTiO3 and describes the correct insulating ferromagnetic ground state with a band gap of approximately 0.7 eV. Our systematic procedure to find the appropriate electron (U) and exchange (J) correlation not only predicts the correct band gap and ground-state magnetic structure but also correctly calculates the energy differences of different magnetic structures, which were found to be quite small
Acknowledgment
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owing to the low Curie temperature of approximately 30 K.
SK (fellow code no. IF140119) acknowledges the Inspire fellowship program of the Department
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Fig. 1. (Color online) (a) Super cell 2 × 2× 2 ABO3 perovskite structure. The B atom is shared by six oxygen atoms forming the octahedron. Because of the deformation of the octahedron, orthorhombic CaVO3 and YTiO3 structures (red boundary) are formed, which are derived from
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the super cell. (b) Orthorhombic ABO3 structure derived from the 2 × 2 × 2 super cell. (c) Cubic and orthorhombic Brillouin zone showing highly symmetric lines. Fig. 2. (Color online) Band structure of (a) cubic SrVO3, (b) orthorhombic CaVO3, and (c) orthorhombic YTiO3 obtained with the generalized gradient approximation. The density of states
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(DOS) and its decomposition into the partial DOS (PDOS) are also shown. Fig. 3.(Color online) (a) Variation of band gap with Ueff for all the 3d1 perovskites. The inset
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shows the narrow region of Ueff for YTiO3. (b) Variation of exchange-correlation energy (J) from 0.1 to 1 eV with the band gap. Fig. 4. (Color online) Electronic structure of orthorhombic YTiO3 calculated by the generalized gradient approximation plus U approach. The density of states (DOS) and its decomposition into the partial DOS (PDOS) are also shown. LHB, lower Hubbard band; UHB, upper Hubbard band. Fig. 5. Total ground-state energies calculated by the generalized gradient approximation plus U approach per unit formula of YTiO3 for ferromagnetic (FM) C-, A-, and G-type antiferromagnetic (AFM) ordering, and paramagnetic (PM) configurations.
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Electronic structure calculations of SrVO3, CaVO3, and YTiO3 by the full-potential linearized augmented plane wave method. Correct ground state with generalized gradient approximation plus U calculations.
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Optimal on-site Coulomb potential predicts the correct band gap and magnetic ground
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state of YTiO3.