We demonstrate here that the correct covalent insulating ground state may be obtained if the antiferromagnetic interaction along the spin chain is included.. This behaviour is commonly r
Trang 1Ground state of spin chain system by Density Functional Theory
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 3 April 2010
Available online 4 May 2010
Keywords:
DFT
Ab initio
Spin chain
Antiferromagnet
Ground state
Ca 2 CuO 3
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 optimi-zation for the periodic structure of Ca2CuO3using 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
1 Introduction
The Cu–O based spin chain system occurs in many compounds,
e.g in A2CuO3(A = Sr, Ca) These perovskite-like compounds
exhi-bit 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
exam-ples, they showed thermally independent covalent insulation state
[2]and spin-charge separation[4] At room temperature, the
com-pounds 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
interac-tion along 1D spin chains and origin of forbidden phonons, which
are coupled with these 1D chains[5–8]
The structure of A2CuO3can be considered in general as a
dou-ble oxide (AO)2(CuO), in which each [–CuO–] layer is inserted
be-tween the other two [–AO–] layers (Fig 1a) There is no extra
oxygen atom at (1/2, 0, 0) position as in the superconducting La
2-CuO4, so the spin-exchange Cu–O network expands only along b
direction, forming a perfect 1D spin structure The Ca2CuO3shows
a small magnetic moment of 0.05lB which originates probably
from the existence of random doublet chains of finite length
(con-taining the odd number of Cu–O units) The estimation for the
in-chain magnetic exchange is about 1200 K and the inin-chain–
interchain exchange ratio is Jk=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 locali-zation of valence band electrons causing the existence of a small gap between the valence and the conduction band From the inves-tigation of photoemission spectra, Maiti et al [9–11]have sug-gested 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 cal-culations 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 A2CuO3is questionable Instead, the known ab initio studies[5,6]argued for a charge-trans-fer 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-hybrid-ization system of Cu 3d and O 2p atomic orbitals Several other the-oretical 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 con-tradicted, as for the metallic band structure in one hand (showed
in Refs.[5,6]) and an insulating ground state in the other (con-firmed 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 0927-0256/$ - see front matter Ó 2010 Elsevier B.V All rights reserved.
* Corresponding author.
E-mail address: namnhat@gmail.com (H.N Nhat).
Contents lists available atScienceDirect
Computational Materials Science
j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c o m m a t s c i
Trang 2spreading Cu 3d electron density over the Fermi level while the O
2p density was almost zeroed in the conduction band (no
pd-hybridization 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)
calcula-tion 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 6-31G basis set used in de Graaf and Illas[6]
showed the band gap of 16 and 1.53 eV respectively, and the
den-sity of state (DOS) that contains only Cu 3d state above the Fermi
level up to 6 eV The reported DOS also exhibits a gap between
0.5 and 2 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
resolu-tion, 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
re-sults have also succeeded in showing the insulating ground state
despite that the antiferromagnetic interaction has been
considered
2 Method
For the systems with strong electron correlation, there are
sev-eral 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
pho-non calculation, the strong electron correlation seemed to have
only limited impacts
The HF level theory, which was used to predict the vibration
fre-quencies in Ref.[7], considers the electron correlation only in a
limited sense, i.e each electron sees and reacts to an averaged
elec-tron density[12] The DFT, in comparison, also accounts for some
interactions of pairs of electrons with opposite spins The DFT di-vides the electron energy into several components which are sep-arately computed: the kinetic energy ET, the electron–nuclear interaction EV, the Coulomb repulsion EJ, and an exchange–correla-tion term EXCwhich accounts for the remainder of electron–elec-tron interaction (which is itself divided into separate exchange and correlation components)[13]:
E ¼ ET
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 fac-tors etc.) The LDA + U approach was also involved to provide the additional insights and comparisons All calculations were per-formed using DMol3package[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 calcula-tion could not easily be performed in the past
3 Geometry optimization and DOS calculation Basically, DMol3[14]utilizes four types of basis sets: the mini-mal basis set (MIN), the double-numeric (DN) basis set (which con-tains approximately two atomic orbitals per one occupied orbital), the DN with diffuse function added (DND) and the DN with polar-ized and diffuse functions added (DNP) The size of the DNP is com-parable to that of the Gaussian type orbital 6-31Gof 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 cut-off of 5.5 Å was used, the k-space equal to 6 10 4 and the Monkhorst–Pack’s k-point sampling scheme were selected as im-plicit The core treatment followed mainly the Effective Core Potentials method For the geometry optimization, the SCF toler-ance was set to 105, the energy convergence 2 105(Ha), max-imal force 0.004 Ha/Å and maxmax-imal distance 0.005 Å Spin-polarized wave functions were used in all calculations
Table 1lists the selected results from the optimization of geom-etry for Ca2CuO3using 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 2shows the DOS as obtained for the periodic model of Ca
2-CuO with optimized geometry The parts (a) and (b) were ob-Fig 1 The periodic structure of Ca 2 CuO 3 (a); the antiferromagnetic supercell ‘1a2b’
(b).
Trang 3tained 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 5 inFig 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 inFig 2b The main
re-sults are as follow: (i) the DOS-s express no observable change
ex-cept 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
contri-bution of the Ca 4s and/or Ca 3p electrons in the area above 4.5 eV
(segments Nos 4, 5 and 6 inFig 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 7 segments, each of which is characterized by a superior
con-tribution 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 2 is clearly resulted from the Cu 3d–O 2p hybridization and the segment No 7 has mainly the O 2s character The area above 0.164 Ha (4.46 eV) which in-cludes the segments Nos 4, 5 and 6, is dominated by the contribu-tion 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 ob-tained DOS-s are basically different than the ones previously re-ported in Refs.[5,6]: even with the use of DND basis set, there was observed no gap spanning between 2.0 and 0.5 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 insuffi-cient treatment for Ca 3d electrons in DND basis set The inclusion
of results calculated with DND basis set inFig 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 6 10 4 To investigate
Table 1
The results of cell optimization.
Parameter PBE/
DND
BLYP/
DND
PBE/
DNP
BLYP/
DNP
Exp (Å) (Ref.
[2] )
Fig 2 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
Trang 4how large is the effect of the Hubbard constraint for d-electron
cor-relation 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 DMol3results The additional settings
in this calculation are: energy cut-off 550 eV, on-the-fly
pseudopo-tentials (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
expli-cit separation U between occupied and unoccupied state of
d-elec-trons 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
conduc-tion 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 Dmol3DOS,
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-elec-tron correlation as the Hubbard model assumes The only
weak-ness 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 Section4
The LDA + U corrected DOS is well reproduced by the
experi-mental 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 CuO4network and the oxygens in the CuO chain admixed with
Cu 3dx2 y 2states 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
distri-bution 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 3.0 and 5.0 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 photoemis-sion in Ref.[11]is of order 0.1 and 0.8 eV respectively, so many fea-tures could not be clearly resolved, e.g the highest O 2p peak at
4.2 eV was overlapped with the 3.0 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 to-gether span across the Fermi level, making the final ground state metallic When the Hubbard constraint U applied, a gap propor-tional 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 A2CuO3by Maiti et al.[11]but the subsequent determination[6]that the O 2p density above the Fermi level was near zero made this consid-eration 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 hybrid-ization 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 3 The schematic energy diagrams for the p–d metal (a), covalent insulator (b), charge-transfer insulator (c) and A 2 CuO 3 (d).
Trang 5the insulating gap (Fig 3c), the experimental studies showed a
covalent insulating mechanism (Fig 3b) We will show in the next
section that for Ca CuO (and A CuO in general) the correct
en-ergy diagram should be as depicted inFig 3d This is a modified covalent insulating scheme with additional Ca 3d components coming from CaO bilayers
Fig 5 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.
Trang 6The large portion of Ca 3d contribution to conduction band
showed in Figs 2d and 5f argues for a possible excitonic
ex-change 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
pro-posed exchange should be non-magnetic charge-transfer
be-tween Cu 3d2z2 and Ca 3d0z2 The illustration for this exchange
mechanism is given inFig 4 However, the detailed description
of exchange parameters tyand tz is a subject for consideration
elsewhere, and we will focus here on how to obtain the correct
covalent insulating ground state
4 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 inFig 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
calcula-tions as reported in Ref.[5] Again, the LDA scheme failed to
pro-duce 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 3d1x2 y 2
elec-trons that occurs right above the Fermi level The previous studies
[5,6]showed that this 1D band corresponds to the band of
antifer-romagnetic exchange along the spin chain We show now that the
correct insulating ground state can be obtained if the
antiferro-magnetic ordering is considered Therefore, we utilized the
antifer-romagnetic 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
par-allel 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 1 (up) and 1 (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
occu-pancy was checked against several smearing factors from 0.005
to 0.0025 and 0.00125 Ha The k-point samplings were tested from
4 3 2 to 6 12 4 (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
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 corre-sponds to the p-antibonding system in the Cu 3d–O 2p pd-hybridization At Fermi level, the d-density falls sharply to zero, and the insulating gap is maintained by the strong electron cor-relation within the pd-hybridization system This situation man-ifests the typical covalent insulating state (Fig 3b) with a well defined Coulomb repulsion between the occupied and non-occu-pied 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 oc-curs just above 1.5 eV To ensure the correct computational re-sults, 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 compar-ison, 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 cut-off 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
5 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 Ca2CuO3is 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
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