Electronic Structure of Weakly Correlated Antiferromagnetic Metal SrCrO3: First-principles calculations

By systematic first-principles calculations, we study the electronic structure and magnetic property of SrCrO$_3$. Our results suggest that SrCrO$_3$ is a weakly correlated antiferromagnetic (AF) metal, a very rare situation in transition-metal oxides. Among various possible AF states, the C-type spin ordering with small amount of orbital polarization (dxy orbital is more occupied than the d_{yz/zx} orbital) is favored. The detailed mechanism to stabilize the C-type AF state is analyzed based on the competition between the itinerant Stoner instability and superexchange, and our results suggest that the magnetic instability rather than the orbital or lattice instabilities plays an important role in this system. The experimentally observed compressive tetragonal distortion can be naturally explained from the C-type AF state. By applying the LDA+$U$ method to study this system, we show that wrong ground state will be obtained if $U$ is large.


I. INTRODUCTION
The electronic and magnetic structures of transition metal perovskites are usually complicated due to the mutual interplay of various degrees of freedom [1][2][3][4] , such as lattice, spin, charge and orbital, in the partially filled d shell. The relative competition or cooperation among those degrees of freedom may stabilize many possible states, which are energetically close. To identify those subtle energy differences among various possible solutions is a hard task and challenging issue for the studies in this field. Nevertheless, the first-principles electronic structure calculations based on density functional theory provides us a possibility to understand the physical trend and many related issues, if the studies are done systematically. Such strategy will be followed in this paper to study the electronic and magnetic properties of SrCrO 3 , a simple perovskite but with many controversial issues.
Limited knowledge is available for SrCrO 3 due to the requirement of high pressure for synthesis. Chamberland 5 first reported SrCrO 3 as Pauli paramagnetic metal with cubic perovskite structure, however in the recent study by Zhou et.al 6 , it is suggested as a cubic paramagnetic insulator with a possible insulator to metal transition under pressure of 4Gpa in their polycrystal samples. Several anomalous properties, such as deviation of Curie-Weiss law of magnetic susceptibility, low Seebeck coefficient, and glassy thermal conductivity, are reported in Zhou's study, and those properties are attributed to the bond-length fluctuation instabilities. On the other hand, a new structural phase, i.e. the perovskite with compressive tetragonal distortion, was reported recently for SrCrO 3 by Attfield et.al 7 . A cubic to tetragonal structure transition was observed around 40K, however no discontinuity of electronic conductivity was observed due to the coexistence of both cubic and tetragonal phases even at low temperature. The relative instability of two structural phases is sensitive to the sample preparation and microstrain. Attfield et.al 7 also suggested that, in the tetragonal phase, the C-type AF state with partial orbital ordering is favored. Alario-Franco et.al 8 again reported SrCrO 3 as a cubic paramagnetic metal, and suggested the 4+ oxidation state of Cr (i.e. Cr 4+ ) by electron energy loss spectroscopy (EELS). No anomaly found in their specific heat measurement for SrCrO 3 9 .
At the theoretical side, K. W. Lee et.al 10  In order to understand the electronic and magnetic properties of SrCrO 3 , in this work, we perform systematic first-principles calculations for this compound. The effects of various degrees of freedom, such as spin, orbital, lattice, and the possible correlation effect are carefully studied step by step. From the calculated results, we conclude that the compressive tetragonal structure with C-type AF spin ordering is the most stable state. A mechanism is proposed to understand the stabilization of such C-type AF metallic state. In contrast to earlier studies, we suggest that the magnetic instability rather than orbital or lattice instabilities plays main role in this system. The controversial issues, such as metal or insulator, cubic or tetragonal, can be naturally clarified based on our systematic analysis.
The methods will be described in section II, and the results and discussions, which are separated into four subsections step by step, will be presented in section III. Finally, a brief summary will be given in section IV.

II. METHOD
The first-principles calculations based on the density functional theory were performed by using the BSTATE (Beijing Simulation Tool for Atom Technology) package 13 , in which the plane-wave pseudopotential method is adopted.  Due to the octahedral local crystal field, the five d orbitals of Cr split into a lower lying three-fold degenerate t 2g manifold and a higher two-fold degenerate e g manifold. The band width of t 2g and e g states are about 2.5eV and 4.2eV respectively. The e g band width is much larger than that of t 2g , because the pdσ-type bonding is stronger than pdπ-type bonding.
The t 2g -e g separation (crystal field splitting) is about 2.4eV, comparable to the t 2g band width. The Fermi level E f lies within the t 2g manifold, while the e g part lies well above E f , suggesting a typical t 2g system. The calculated total occupation number of Cr-3d orbitals is n = 1.994, quite consistent with chemical stoichiometry analysis and the EELS experiment which suggests the Cr 4+ valence state by Alario-Franco 9 .
All the three t 2g bands cross the Fermi Level, forming three sheets of Fermi surface (FS), as shown in Fig.3. There is also a large section of flat band structure along Γ(0, 0, 0) − X(0, π, 0) − M(π, π, 0) high symmetry lines, which leads to the sharp DOS peak located the display for total three sheets. The Γ-point locates at the center of the cube.
at 0.5eV above E f . The existence of such Van-Hove singularity will affect the electronic structures considerably, as will be addressed below. Among the three sheets of FS, two of them are cubic like with flat facets, suggesting the possible existence of FS nesting 10 . In order to clarify this point, we calculated the Lindhard response function χ 0 (q), as shown in Fig.4.
There are a plateau locating around q = (π, π, 0) and a sharp peak locating at q = (π, π, π), with the latter peak slightly higher than the former. The presence of two structures in the response function corresponds to two kind of instabilities, which are related to the C-type and G-type AF ordering respectively, as will be discussed below. Considering the fact that the peak at q = (π, π, π) is higher than that at q = (π, π, 0), it may be expected that the expectation is not consistent with our following calculations.

B. Magnetic Instability in Cubic Structure
Now we start to include the spin degree of freedom into our studies. For this purpose, we neglect the possible lattice distortion, and only concentrate on the cubic structure. We calculate the total energies of different magnetic states as function of volume. As shown in Fig.5, the C-type AF state is always stabilized for all the calculated volumes. In other words, the C-type AF state is the most stable state even without any lattice distortion. This    is one of our main conclusions, which will be carefully discussed in this part. The calculated moment are 1.63µ B /Cr, 1.59µ B /Cr, 1.56µ B /Cr, 1.55µ B /Cr for FM, A-type, C-type, G-type states respectively. For the FM state, the magnetic moment quickly collapses with volume compression as show in Fig.6, but for the AF states (A-type, C-type and Gtype), the moments are not so sensitive to the volume change. This again suggests that although the system is close to itinerant Stoner instability, the stabilization of AF states is beyond the Stoner mechanism.
Although we only treat the cubic structure, the A-type and C-type magnetic ordering will further break the symmetry, and lead to orbital ordering. As shown in Fig.7-10 and listed in Table I, the degeneracy between the d xy and the d yz/zx orbitals is lifted in the presence of A-type or C-type AF states. More importantly, the A-type and C-type AF states lead to different orbital ordering: the occupation of d xy orbital is larger (smaller) than d yz/zx orbitals for the C-type (A-type) AF state. Recall the fact that current calculations are based on cubic structure, the observed orbital ordering must be purely due to the effect of magnetic orderings. If we evaluate the orbital polarization as the occupation number difference between the d xy and the d yz/zx orbitals, defined as n p = n dxy −n d yz/zx , the calculated values suggest that the polarization is about n p = −0.04 and n p = 0.06 electrons for the A-type and C-type AF state respectively. Now we have to answer several important questions: (1) why the types of orbital ordering are opposite for A-type and C-type AF states? (2) why the C-type AF state is more stable than the A-type AF state? To answer these questions, the detailed understanding to the superexchange is important. Through the superexchange mechanism, the system gains energy from hybridizations: the hybridization between the occupied states (at one site) and the unoccupied states (at neighboring sites) will lower the energy of the occupied states by approximately t 2 /∆, where t is an effective transfer integral and ∆ is an energy difference between occupied and unoccupied states. We start from the C-type AF state in which the spins of two adjacent Cr 4+ ions lie antiparallely in ab-plane but parallely along c-axis. The occupied majority spin d xy (d yz/zx ) orbital is pushed down by hybridization with unoccupied majority spin d xy (d yz/zx ) orbital located at nearest-neighboring Cr site in ab-plane (The effect is forbidden along the c-axis due to the parallel spin chain). Such hybridization process can happen four times for d xy orbital, but only two times for d yz/zx orbital due to its spacial orientation. Therefore, the occupied majority spin d xy orbital is pushed down more strongly than the d yz/zx orbitals due to the larger hybridization. Applying the same analysis to the A-type AF state in which the spins of two adjacent Cr 4+ ions lie parallely in ab-plane but antiparallely along c-axis, only the hybridization paths along the c-axis need to be considered. Now, for the A-type AF state, the hybridization process can happen two times for d yz/zx orbital, but zero time for d xy orbital again due to its spacial orientation. Therefore, the d yz/zx state is lower than the d xy state for the A-type AF state, in opposite to the situation in C-type AF state. This will answer the first question listed above. Due to the same reason, the system can gain energy from eight possible hybridization paths in the C-type AF state, but only four paths in A-type AF state. This is the reason why the C-type AF state is more stable than A-type AF state.
According to the above mechanism, it is immediately realized that the G-type AF state should be more stable than A-type or C-type AF states. Why it is not the case from the first principle calculations? The reason is following: applying the superexchange mechanism in the above discussion, we assume that the majority spin d xy or d yz/zx orbitals are fully occupied. This is of course not the case, due to the large band width and the small splitting between these t 2g orbitals, the system is always metallic and these t 2g states are partially occupied. In this situation, Stoner mechanism, which favors itinerant ferromagnetism due to the kinetic energy gain, also plays its role, and finally the full AF state (i.e, the G-type AF state) lost its energy gain. The system prefers to keep certain FM spin orientation along particular directions, such as c-axis in C-type and ab-plane in A-type AF state. Therefore, the stabilization of C-type AF state comes from the subtle balance between the itinerant Stoner mechanism and the superexchange.
To further prove our above discussion, we performed the following calculations. Keeping the cubic lattice structure, we calculate the energy difference between various magnetic state as function of number of electrons. The virtual crystal approximation(VCA) 16 is used here to simplify our studies. For the parent compound, the total number of t 2g electron on Cr 4+ is 2. By increasing the number of d electrons, the majority spin t 2g states tend to be fully occupied. As we can see from Fig.11, if we increase the number of electrons, indeed the G-type AF state is stabilized.

C. Effect of Lattice Distortion
In our above analysis, only the cubic structure is treated. However, the tetragonally distorted structure is also observed experimentally, the possible tetragonal distortion and its effect on electronic structure must be studied. The tetragonal phase must be energetically very close to the cubic phase, duo to the reported coexistence of these two phases at low temperature 7 . Therefore, in this section, we try to understand the effect of tetragonal structure distortion. The main conclusions drew from this section are following: (1) If tetragonal distortion is allowed, the compression (rather than elongation) along the c-axis should be favored, in consistent with experimental observation. (2) C-type AF state and corresponding  orbital ordering are further stabilized by compressive tetragonal distortion. (3) The energy gain due to tetragonal distortion (about 10meV/Cr) is smaller than that coming from the superexchange (about 100meV/Cr) discussed above. (4) The orbital polarization induced by tetragonal distortion is much smaller than that caused by magnetic orderings. Fig.12 shows the optimized total energy as function of volume after allowing the tetragonal distortion. Comparing with the cubic results shown in Fig.5 13: The optimized c/a ratio for A-type and C-type AF state at different volume. Volume compression will suppress the lattice distortion. V 0 is the experimental volume.
favors elongation distortion along the c-axis as shown in Fig.13. For the most stable C-type AF state, the optimized c/a ratio is 0.990, in good agreement with the experimentally observed tetragonal distortion (c/a=0.992). Such an opposite tendency for A-type and C-type AF states can be easily understood from the orbital polarization discussed above. In the C-type AF state, the d xy orbital is more occupied than the d yz/zx orbitals, therefore c-axis compression is favored due to the reduction of electron static potential. The opposite is true for A-type AF state. As we expected, the lattice distortion will further enhance the orbital polarization, and stabilize the A or C-type AF states. In reality, A-type and C-type AF states are further stabilized by only 2.0meV/Cr and 6.0meV/Cr respectively, comparing with their cubic phase. On the other hand, the FM and G-type AF states have no energy gain by lattice distortion.
Another important issue is the origin of orbital polarization observed both in experiment and LDA calculations. Since both lattice distortion (due to crystal field) and AF magnetic ordering due to superexchange mechanism can lead to orbital polarization, we have to determine which one is the dominate factor. From our LDA calculations shown in table I, it is suggested that the magnetic ordering should be the dominant factor. First of all, the energy gain due to the lattice distortion is small (less than 10meV), even smaller than the energy difference among the A-type, C-type, G-type AF states (about20 ∼ 50meV ) calculated in the cubic phase. Second, as shown in table I, where the calculated orbital polarization of various magnetic states for different c/a ratio are listed, the orbital polarization coming from the lattice distortion is smaller than that coming from the superexchange effect calculated in the cubic phase.
Finally, we conclude that although the tetragonal distortion is favored, the distorted phase is energetically close to the cubic phase, in consistent with experimental results of coexistence of the two phases. On the other hand, an electronic structure transition reported experimentally around 4 GPa 6 is not observed from our calculations.

D. The Effect of On-site Coulomb Correlation
One important issue in transition metal compounds is the possible strong correlation, which can not be treated successfully by LDA level calculations. We therefore need to check the effect of electron correlation by advanced technique. Given the fact that SrCrO 3 should be more itinerant than CaCrO 3 due to the lager band width, and also the fact that most of the electron properties calculated above using LDA for SrCrO 3 can be well compared with experiment, we expect that the effect of on-site Coulomb U is not significant. Nevertheless, we will show, in this part by LDA+U calculations, that the SrCrO 3 system is indeed not strongly correlated system. In principle, LDA+U is a cheap technique, which is suitable for strongly correlated AF insulating state (large U limit), however we will show that the LDA+U results for SrCrO 3 are nonphysical if large U is applied. Fig.14 shows the calculated total energy relative to FM state as function of U. We can see clearly that the A-type AF state is stabilized at large U limit (U > 4.0eV), which is not consistent with experiment. What is more surprising is that, for the C-type AF state, an orbital ordering transition is observed, as shown in Fig.15. For the small U limit, the d xy orbital is more populated than d yz/zx orbitals, however for the large U limit, the d yz/zx orbital become more occupied, in opposite to the small U limit. The appearance of this transition can be understood as the following. In the high spin configuration of Cr 4+ , the two 3d electrons will occupy the t 2g orbitals, which split into one d xy orbital and two degenerate d yz/zx orbital in the C-type AF state. If the d xy orbital is lower in energy, the system has to be metallic due to the degeneracy between the d yz and the d zx orbitals. However, if the U is very large (large than band width), it will try to reverse the energy level ordering of d xy and d yz /d zx states, in such a way that two d yz/zx orbitals are mostly occupied and d xy becomes nearly empty, then the system gains energy from the possible gap opening between the occupied d yz/zx and unoccupied d xy orbitals if U is large.
The above analysis for orbital polarization suggest that, no matter the system is ordered in the C-type or the A-type AF states, the d yz/zx orbitals are always more favored than the d xy orbital in the large U limit. In such an orbital polarization, the elongation (rather than compressive) tetragonal distortion should be favored, in opposite to experimental observation. Therefore, our calculations support the conclusion that the SrCrO 3 is at least not strongly correlated system. Furthermore, from above studies, we also learn that the metallicity of SrCrO 3 in the C-type AF state is protected by the degeneracy between the d yz and the d zx orbitals given the compressive tetragonal distortion observed. From the very good agreement between the optimized c/a ratio from LDA calculation (c/a=0.990) and that observed experimentally (c/a=0.992), we can further conclude that the effect of correlation is in fact very weak.

IV. SUMMARY
In summary, we study the electronic structure and the magnetic property of SrCrO 3 systematically based on the LDA and LDA+U calculations. We analyze the various magnetic instabilities, the effect of lattice distortion, and the effect of correlation. The controversial issues, such as metal or insulator, cubic or tetragonal, can be now understood in a consistent picture. Our calculated results are also quite consistent with available experiment results.
We finally conclude that SrCrO 3 is a weakly correlated AF metal with small amount of orbital polarization. The magnetic instability rather than the orbital or lattice instabilities plays the dominant role in this compound.

V. ACKNOWLEDGMENTS
We acknowledge the valuable discussions with H. J. Zhang, X. Y. Deng, and the supports from the NSF of China, the 973 Program of China, and the International Science and