Computational Materials Science 49 (2010) S348–S354 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci Ground state of spin chain system by Density Functional Theory Hoang Nam Nhat *, Nguyen Thuy Trang Faculty of Physics and Nanotechnology, UET, Vietnam National University Hanoi, 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam a r t i c l e i n f o Article history: Received 11 October 2009 Received in revised form 24 February 2010 Accepted April 2010 Available online May 2010 Keywords: DFT Ab initio Spin chain Antiferromagnet Ground state Ca2CuO3 a b s t r a c t The Cu–O based spin chain system A2CuO3 (A = Sr, Ca) has attracted considerable attention of scientists during the last decades due to its unique electronic structure This paper presents a ground state optimization for the periodic structure of Ca2CuO3 using the Density Functional Theory (DFT) The electronic structure analysis was carried out on the basis of cell optimization using the PBE functional with DNP basis set Until now, the numerical results were obtained only with exclusion of CaO bilayers from full ab initio treatment We demonstrate here that the correct covalent insulating ground state may be obtained if the antiferromagnetic interaction along the spin chain is included The best estimated band gap is 0.79 eV We also show that the CaO bilayers sufficiently contributed to the distribution of d-states above Fermi level Ó 2010 Elsevier B.V All rights reserved Introduction The Cu–O based spin chain system occurs in many compounds, e.g in A2CuO3 (A = Sr, Ca) These perovskite-like compounds exhibit a strong spin ½ antiferromagnetic coupling between the copper atoms along the Cu–O chains and offer themselves as the perfect examples of one-dimensional (1D) antiferromagnets [1] The lack of dimensionality causes many interesting behaviours, for examples, they showed thermally independent covalent insulation state [2] and spin-charge separation [4] At room temperature, the compounds exhibited extremely short optical excitation life-time and colossal dielectric constant (>10,000) [2,3] The observation of the spin-charge separation is one of novelties which might have substantial impact on future of quantum devices Various studies were performed to explain their strong antiferromagnetic interaction along 1D spin chains and origin of forbidden phonons, which are coupled with these 1D chains [5–8] The structure of A2CuO3 can be considered in general as a double oxide (AO)2(CuO), in which each [–CuO–] layer is inserted between the other two [–AO–] layers (Fig 1a) There is no extra oxygen atom at (1/2, 0, 0) position as in the superconducting La2CuO4, so the spin-exchange Cu–O network expands only along b direction, forming a perfect 1D spin structure The Ca2CuO3 shows a small magnetic moment of 0.05 lB which originates probably from the existence of random doublet chains of finite length (containing the odd number of Cu–O units) The estimation for the inchain magnetic exchange is about 1200 K and the inchain– * Corresponding author E-mail address: namnhat@gmail.com (H.N Nhat) 0927-0256/$ - see front matter Ó 2010 Elsevier B.V All rights reserved doi:10.1016/j.commatsci.2010.04.002 interchain exchange ratio is J k =J ? % 300 [5,6] This compound also shows a very high resistivity at room temperature (of order several hundreds MX) while its activation energy is as small as of tens meV [2,3] This behaviour is commonly referred to as the covalent insulation state which is believed to be associated with the localization of valence band electrons causing the existence of a small gap between the valence and the conduction band From the investigation of photoemission spectra, Maiti et al [9–11] have suggested that there was an overlapping of the upper Hubbard band and the O 2p-derived bands above Fermi level, so the covalent insulation state appears as a result of a strong electron correlation in many-body (non-relativistic) quantum system The analysis of Maiti et al., however, did not find much support from ab initio calculations of electronic structure performed by various groups [5,6] As there was observed only a trace of O 2p density above the Fermi level, the covalent insulation behaviour of A2CuO3 is questionable Instead, the known ab initio studies [5,6] argued for a charge-transfer insulating ground state They showed that the insulating gap had pure charge-transfer character between the Cu 3d bands and was not a result of strong electron repulsion inside the pd-hybridization system of Cu 3d and O 2p atomic orbitals Several other theoretical studies are also available for the analysis of optical phonons in the doped and un-doped Ca2CuO3 [7,8] as there are continuous interests for the optical manipulation of 1D spin chains Many aspects of the ground state still remain unknown or contradicted, as for the metallic band structure in one hand (showed in Refs [5,6]) and an insulating ground state in the other (confirmed in experimental studies [2,3,9–11]) While the strong onsite Coulomb repulsion between occupied and unoccupied d-electrons along the 1D spin chain is expected, the calculation showed a H.N Nhat, N.T Trang / Computational Materials Science 49 (2010) S348–S354 S349 interactions of pairs of electrons with opposite spins The DFT divides the electron energy into several components which are separately computed: the kinetic energy ET, the electron–nuclear interaction EV, the Coulomb repulsion EJ, and an exchange–correlation term EXC which accounts for the remainder of electron–electron interaction (which is itself divided into separate exchange and correlation components) [13]: E ¼ ET þ EV þ EJ þ EXC Fig The periodic structure of Ca2CuO3 (a); the antiferromagnetic supercell ‘1a2b’ (b) spreading Cu 3d electron density over the Fermi level while the O 2p density was almost zeroed in the conduction band (no pdhybridization above Fermi level) [5,6] The main results from the previous studies may be summarized as follows: (i) the LDA (with minimum basis sets, i.e Cu-(4s, 4p, 3d), O-(2s, 2p), Sr-(5s, 5p, 4d), Ca-(4s, 4p, 3d)) and the LMTO (Linear Muffin-Tin Orbital) calculation by Rosner et al [5] showed the metallic band structure and the band gap of 2.2 and 2.0 eV for the two models correspondingly; (ii) the UHF and B3LYP calculation with the Gaussian type basis set of size similar to that of 6-31Gà basis set used in de Graaf and Illas [6] showed the band gap of 16 and 1.53 eV respectively, and the density of state (DOS) that contains only Cu 3d state above the Fermi level up to eV The reported DOS also exhibits a gap between À0.5 and À2 eV which was explained as originated in a possible failure of B3LYP hybridization scheme in the systems with strong electron correlation such as in A2CuO3 [6] It is worth to note that the previously reported DOS-s appeared somehow in low resolution, which is apparently associated with the use of small basis sets, k vectors, or in this particular case, with the treatment of CaO bilayers only as the static potential layers (therefore excluding their contribution to the total density of states) The mentioned studies have also not included the structure optimization as part of their electronic structure calculation, so there was not clear a consistency between the established electronic structures and the experimental geometries of A2CuO3 No previous ab initio results have also succeeded in showing the insulating ground state despite that the antiferromagnetic interaction has been considered Method For the systems with strong electron correlation, there are several ways to account for this collective effect Some amount of the electron correlation is already present in the Hartree–Fock (HF) approximation but higher level theories such as CIS, MP2 and DFT treat the electron correlation more extensively For A2CuO3, the known DFT studies are from de Graff and Illas [6], and Hoang et al [8] The obtained results in Ref [8] showed that for the phonon calculation, the strong electron correlation seemed to have only limited impacts The HF level theory, which was used to predict the vibration frequencies in Ref [7], considers the electron correlation only in a limited sense, i.e each electron sees and reacts to an averaged electron density [12] The DFT, in comparison, also accounts for some ð1Þ Each of these parts is a function of electron density All of them correspond to the classical energies of charge distribution except the last EXC This component arises from the anti-symmetry of the quantum mechanical wave-function of electron (exchange functional EX), and the dynamic correlation in the motions of the individual electrons (correlation functional EC) With this inclusion, the DFT method achieves a significantly greater accuracy than the HF method at only modest increase in computation time The computational strategy used in this work is as follows (i) We applied the full ab initio treatment for all atoms including the Ca atoms, i.e the CaO bilayers were included in the calculation of density of states, the partial densities were also computed for Ca (ii) Before the electronic structure calculation, the cell structure was optimized by various functionals and basis sets, starting from the experimental values for Ca2CuO3 (a = 3.254, b = 3.778, c = 12.235 Å, space group Immm) The most consistent functional/ basis set configuration was then selected on the basis of the best fit (iii) The DOS-s were computed with various k vectors, including the large and very large values, and at different orbital qualities (cut-off in dimension and energy, density mixing and smearing factors etc.) The LDA + U approach was also involved to provide the additional insights and comparisons All calculations were performed using DMol3 package [14] except for LDA + U calculations which were done by CASTEP code [15] In comparison with the studies previously reported, the calculation according to the given plan is much more time-consumed, especially because of applying large basis sets to CaO bilayers and of expanding the basis sets along large k spaces It is quite apprehensive that such a calculation could not easily be performed in the past Geometry optimization and DOS calculation Basically, DMol3 [14] utilizes four types of basis sets: the minimal basis set (MIN), the double-numeric (DN) basis set (which contains approximately two atomic orbitals per one occupied orbital), the DN with diffuse function added (DND) and the DN with polarized and diffuse functions added (DNP) The size of the DNP is comparable to that of the Gaussian type orbital 6-31GÃà of Hehre et al [16] but the DNP performs much quicker and is also more accurate For our calculation, where not specifically given, a real space cutoff of 5.5 Å was used, the k-space equal to  10  and the Monkhorst–Pack’s k-point sampling scheme were selected as implicit The core treatment followed mainly the Effective Core Potentials method For the geometry optimization, the SCF tolerance was set to 10À5, the energy convergence  10À5 (Ha), maximal force 0.004 Ha/Å and maximal distance 0.005 Å Spinpolarized wave functions were used in all calculations Table lists the selected results from the optimization of geometry for Ca2CuO3 using two functionals BLYP and PBE applied on the DND and DNP basis sets It is apparent from the data given that the DND basis set for both functionals has resulted in the larger cells The most consistency (with differences less than 1.5%) is seen for the PBE/DNP setting, so this configuration was chosen for the next calculations Fig shows the DOS as obtained for the periodic model of Ca2CuO3 with optimized geometry The parts (a) and (b) were ob- S350 H.N Nhat, N.T Trang / Computational Materials Science 49 (2010) S348–S354 Table The results of cell optimization Parameter PBE/ DND BLYP/ DND PBE/ DNP BLYP/ DNP Exp (Å) (Ref [2]) a b c 3.435 3.871 12.745 3.459 3.908 12.847 3.300 3.818 12.332 3.334 3.853 12.459 3.254 3.778 12.235 tained using the DND basis set (the reason for including the DND basis set in this figure will be clear during the discussion) The curve No in Fig 2a is due to the replacement of PBE functional by the BLYP one As depicted, the replacement seemed to have a negligible impact on the appearance of DOS The analysis of partial DOS-s for a largest k (20  30  10) is given in Fig 2b The main results are as follow: (i) the DOS-s express no observable change except for a very large k value; (ii) the Ca 3d partial DOS above Fermi level is everywhere 0; (iii) the large k revealed a sufficient contribution of the Ca 4s and/or Ca 3p electrons in the area above 4.5 eV (segments Nos 4, and in Fig 2b) As the analysis of the partial DOS-s shows, the extension above 0.4 Ha corresponds mainly to the Ca 3p and Cu 4p states In general, the total DOS can be divided into segments, each of which is characterized by a superior contribution from one type of atomic orbitals The segment No 1, which contains a peak at 0.143 Ha (3.9 eV), corresponds to the density of Cu 4s, Cu 4p, Ca 4s and O 2p electrons A closest Cu 3d, O 2p peak occurs at 0.0215 Ha (0.58 eV) which encloses a gap of 3.3 eV Recall that the Cu 3d–Cu 4s coupling was observed near 0.2e (experimentally reported in Ref [17]) The calculated integral density of Cu 3d electrons from a peak in segment No 1, which can contribute to this Cu 3d–Cu 4s coupling, is about 0.56e ($2.3% of the total d-density) The segment No is clearly resulted from the Cu 3d–O 2p hybridization and the segment No has mainly the O 2s character The area above 0.164 Ha (4.46 eV) which includes the segments Nos 4, and 6, is dominated by the contribution from Ca 3p and Cu 4p orbitals A large gap (0.325 Ha, or 8.5 eV) is seen between the core O 2s level and the valence band The obtained DOS-s are basically different than the ones previously reported in Refs [5,6]: even with the use of DND basis set, there was observed no gap spanning between À2.0 and À0.5 eV and above the Fermi level Furthermore, there is one important point with DND basis set: the absence of the partial Ca 3d contribution to the total DOS This situation was caused mainly by the insufficient treatment for Ca 3d electrons in DND basis set The inclusion of results calculated with DND basis set in Fig 2a and b shows that the smaller basis sets can not reveal the Ca 3d contribution above Fermi level One should applied DNP or larger basis sets to be able to retrieve the positive non-occupied Ca 3d densities From the analysis given, we adapted for the next calculations the optimal k vector spacing equal to  10  To investigate Fig The total density of states (DOS) for the periodic model (a) and the atomic orbital projected partial DOS (b) as obtained by using the DND basis set Note that for the DND setting, the Ca 3d density is everywhere zeroed The effect of the inclusion of an explicit Hubbard U value of 3.5 eV is demonstrated in (c) and the analysis of the partial contributions to the total DOS for the DNP basis set is shown in (d) H.N Nhat, N.T Trang / Computational Materials Science 49 (2010) S348–S354 S351 Fig The schematic energy diagrams for the p–d metal (a), covalent insulator (b), charge-transfer insulator (c) and A2CuO3 (d) how large is the effect of the Hubbard constraint for d-electron correlation on the final distribution of states, we performed the LDA + U calculation for various U values from 1.8 to 7.5 eV Fig 2c compares the results as obtained with and without LDA + U (3.5 eV) using CASTEP code [15] Since CASTEP utilizes the plane wave basis and manages resolution by energy cut-off instead of dimension cut-off (as in DMol3), the CASTEP results would provide the valuable comparison to DMol3 results The additional settings in this calculation are: energy cut-off 550 eV, on-the-fly pseudopotentials (O 550, Ca 280, Cu 400 eV) As seen, the extension about 0.6 eV on each side of DOS appeared when the LDA + U (3.5 eV) constraint was used Reasonably, this shift was caused by the explicit separation U between occupied and unoccupied state of d-electrons Visually, the total DOS and the partial d-DOS look very similar to each other which signifies that the contribution from d-electrons plays a key role in shaping of both valence and conduction bands This result has not been observed before and appears in sharp contrast to almost blank conduction band (only small Cu 3d density above Fermi level) as reported in Ref [6] The existence of a large partial d-density contribution above Fermi level certainly comes from the CaO bilayers It is evident that no such observation could be made in the studies, which excluded the CaO bilayers from the full ab initio treatment Fig 2d demonstrates that the similar DOS can be obtained using the DNP basis set (polarized function added) in DMol3 Here we may clearly observe that except the O 2s part of the Dmol3 DOS, which extends a little below the LDA + U DOS ($ 0.2 eV), all other features appear almost the same Therefore, the use of larger wave function basis sets with PBE functional produced the same d-electron correlation as the Hubbard model assumes The only weakness of this result is the presence of Cu 3d density at Fermi level and the corresponding band structure (Fig 5) is metallic (will be discussed in Section 4) The LDA + U corrected DOS is well reproduced by the experimental data reported in Ref [11] For the features above the Fermi level, the authors in Ref [11] argued that the peak at 3.4 eV is attributed to the O 2p states arising from the apical oxygens of the CuO4 network and the oxygens in the CuO chain admixed with Cu 3dx2 Ày2 states Our results, however, showed that although some small portions of the O 2p states exist at 3.4 eV and above, almost all O 2p states reside below 1.5 eV and the feature at 3.4 eV mainly corresponds to the unoccupied Ca 3d levels This means that above 1.5 eV the transition induced by the excitation in the 1D spin chain Cu–O to the Cu 4s, Cu 4p levels plays only a minor role The distribution of states above 1.5 eV is almost solely controlled by the CaO bilayers Below the Fermi level, the photoemission spectra as re- ported in Ref [11] show two peaks near À3.0 and À5.0 eV which correspond correctly to the Cu 3d and O 2p features in the DOS-s (highest and next to highest d-electron density) Note that the experimental resolution for the ultraviolet and X-ray photoemission in Ref [11] is of order 0.1 and 0.8 eV respectively, so many features could not be clearly resolved, e.g the highest O 2p peak at À4.2 eV was overlapped with the À3.0 peak Fig 3a shows the typical energy diagram for the p–d metals where the conducting O 2p band overlaps the Cu 3d band and together span across the Fermi level, making the final ground state metallic When the Hubbard constraint U applied, a gap proportional to U would separate the valence band from the conduction band, so introducing a state so-called the covalent insulation (Fig 3b) This scenario was originally proposed for the A2CuO3 by Maiti et al [11] but the subsequent determination [6] that the O 2p density above the Fermi level was near zero made this consideration fragile Our calculation showed that Maiti et al [11] were correct, as there was a positive Cu 3d–O 2p pd-hybridization at Fermi level (Fig 3d) The only remaining thing is that this hybridization spans across the Fermi level so the compound is predicted to be a metal instead of an insulator Indeed, the compound is itself an insulator at room temperature with the resistivity reached above 108 X A small gap of 0.19 eV, however, was reported on the basis of resistivity measurement [2] The debates continue until the present on what insulating model is appropriate for A2CuO3 While the calculations resulted in a charge-transfer character of Fig The magnetic and non-magnetic exchange mechanisms in A2CuO3 S352 H.N Nhat, N.T Trang / Computational Materials Science 49 (2010) S348–S354 Fig The band structures as obtained from LDA + U (3.5 eV) and PBE/DNP settings using the original cell (a and b) and ‘1a2b’ antiferromagnetic supercell (c and d) The antiferromagnetic DOS at various smearing factors (e) and partial DOS (f) as obtained with PBE/DNP setting the insulating gap (Fig 3c), the experimental studies showed a covalent insulating mechanism (Fig 3b) We will show in the next section that for Ca2CuO3 (and A2CuO3 in general) the correct en- ergy diagram should be as depicted in Fig 3d This is a modified covalent insulating scheme with additional Ca 3d components coming from CaO bilayers H.N Nhat, N.T Trang / Computational Materials Science 49 (2010) S348–S354 The large portion of Ca 3d contribution to conduction band showed in Figs 2d and 5f argues for a possible excitonic exchange between the occupied Cu 3d states and the unoccupied Ca 3d states If this would happen, then it would be a kind of interlayer (CaO)–(CuO) 3d electron transfer taken in the vertical direction along axis c This would represent a Coulomb repulsion between the 1D Cu–O spin chains and CaO bilayers and the proposed exchange should be non-magnetic charge-transfer be2 tween Cu 3dz2 and Ca 3dz2 The illustration for this exchange mechanism is given in Fig However, the detailed description of exchange parameters ty and tz is a subject for consideration elsewhere, and we will focus here on how to obtain the correct covalent insulating ground state Antiferromagnetic insulating ground state We now turn to the band structures as obtained from the PBE/ DNP and LDA + U calculations The results are given in Fig 5a and b The band structures appear very similar to each other, and agree in general with the ones obtained from LDA–LCAO and LMTO calculations as reported in Ref [5] Again, the LDA scheme failed to produce the insulating ground state as we may see in the given band structures a clear metallic character with a well defined 1D band running along the b direction The width of this 1D band is 1.9 eV for DNP and 2.2 eV for LDA + U settings The similar values were reported in Ref [5], e.g 2.2 eV for the LDA–LCAO and 2.0 eV the LMTO method As seen, the PBE/DNP calculation has lowered the upper bound of this 1D band closer to the zero level For the band gap, we have obtained 1.23 eV for DNP and 1.27 eV for LDA + U Recall that the band gap determined in Ref [6] by UHF is 16 eV (physically meaningless), and by B3LYP is 1.53 eV The experimental optical gap is around 1.7 eV By comparison of the position of 1D band with the corresponding DOS, we observed that this band is closely related to the small density of Cu 3dx2 Ày2 electrons that occurs right above the Fermi level The previous studies [5,6] showed that this 1D band corresponds to the band of antiferromagnetic exchange along the spin chain We show now that the correct insulating ground state can be obtained if the antiferromagnetic ordering is considered Therefore, we utilized the antiferromagnetic models (AF models) (Fig 1b) These models consist of larger unit cell with doubled b-axis (denoted ‘1a2b’ cell; this is to allow the modeling of antiferromagnetic spin exchange between two Cu atoms in the spin chain) The symmetry of a supercell was treated as P1 In this configuration we must consider two parallel Cu–O chains, one starts with Cu atom at (0, 0, 0) (chain 1) and another with Cu at (1/2, 1/4, 1/2) (chain 2) The first Cu atoms of the two spin chains may show ferromagnetic or antiferromagnetic ordering, and depending on this, we distinguished between AF1 (ferro) and AF2 (antiferro) models For AF1, the formal spin states of Cu atoms: Cu1 and Cu2 (chain 1), Cu1’ and Cu2’ (chain 2) were set as (up) and À1 (down) respectively These starting values were then optimized, keeping their equality constrained, during the search for ground state Theoretically, these two models should give the same result as two spin chains are equivalent in original structure with full symmetry (Immm) The test was carried out for various settings of (charge, spin) in mixing of density (ranging from (0.2, 0.5) to (0.1, 0.25) and (0.05, 0.125)) The orbital occupancy was checked against several smearing factors from 0.005 to 0.0025 and 0.00125 Ha The k-point samplings were tested from   to  12  (i.e from 12 to 144 k-points in Monkhorst– Pack mesh) The insulating ground state was found with a well defined band gap which varied a little from 0.73 to 0.79 eV above the changing of smearing factors (Fig 5c and d) The calculated band gaps are quite larger than the activation energy 0.19 eV but are S353 still far below the optical gap 1.70 eV The obtained DOS (Fig 5e and f) showed that there are overlapping positive densities (nearly of equal magnitude) for Cu 3d and O 2p electrons which peak out roughly at 1.0 eV above Fermi level This feature corresponds to the p-antibonding system in the Cu 3d–O 2p pdhybridization At Fermi level, the d-density falls sharply to zero, and the insulating gap is maintained by the strong electron correlation within the pd-hybridization system This situation manifests the typical covalent insulating state (Fig 3b) with a well defined Coulomb repulsion between the occupied and non-occupied d-states There is also a smaller but observable amount of Ca 3d density at 1.0 eV but the main portion of Ca 3d state occurs just above 1.5 eV To ensure the correct computational results, we also performed the additional searches for the 1a2b cell with the unconstrained and undefined starting formal spins The band structures obtained were always metallic For comparison, the calculation was also done with LDA + U functional (U for Cu 3d increased from 2.5 to 7.0 eV with step 0.25 eV) using CASTEP code Settings for CASTEP were as follow: energy cutoff 550 eV, on-the-fly potentials, Pulay density mixing scheme The calculation showed the physically meaningful results only for 2.5 < U < 5.0 eV with maximum insulating gap 0.5 eV at U = 4.75 eV Both LDA + U and PBE functionals underestimated the experimental optical gap 1.70 eV Conclusion We have shown that the DFT with PBE functional and DNP basis set could successfully lead to a correct insulating ground state for Ca2CuO3 Although the DFT underestimated the insulating gap, it provided a good explanation for the covalent insulating regime in this compound The results obtained proposed that the band gap in Ca2CuO3 is maintained by a strong Coulomb repulsion between the occupied and unoccupied Cu 3d electrons inside the Cu 3d–O 2p pd-hybridization system The inclusion of CaO bilayers in the full ab initio treatment also revealed a sufficient contribution of Ca 3d state to total density of state above Fermi level Acknowledgment The authors would like to thank the financial support from the Project QGTD-0904 from Vietnam National University, Hanoi, 2009-2011 References [1] T Ami, M.K Crawford, R.L Harlow, Z.R Wang, D.C Johnston, Q 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chains, one starts with Cu atom at (0, 0, 0) (chain 1) and another with Cu at (1/2, 1/4, 1/2) (chain 2) The first Cu atoms of. .. 2p states reside below 1.5 eV and the feature at 3.4 eV mainly corresponds to the unoccupied Ca 3d levels This means that above 1.5 eV the transition induced by the excitation in the 1D spin chain