Received 3 March 2008

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Study on the ground state of NiO: The LSDA (GGA)+U method

Tuo Cai a, Huilei Han b, You Yu a, Tao Gao a,, Jiguang Du a, Lianghuan Hao a

a Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, Chinab College of Mathematics, Sichuan University, Chengdu 610064, China

a r t i c l e i n f o

Article history:

Received 3 March 2008

Received in revised form

4 October 2008

Accepted 9 October 2008

PACS:

71.15.Ap

71.20.b

71.27.+a

Keywords:

NiO

FLAPW

GGA+U

LSDA+U

a b s t r a c t

The ground-state properties of NiO have been investigated using the all-electron full-potential

linearized augmented plane wave (FLAPW) and the so-called LSDA (GGA)+U (LSDAlocal-spin-density

approximation; GGAgeneralized gradient approximation) method. The calculated result indicates that

our estimation ofUis in good agreement with experimental data. It is also found that none of the L SDA

(GGA) methods is able to provide, at the same time, accurate electronic and structural properties of NiO.

Although the GGA+U method can properly predict the electronic band gap, it overestimates the lattice

constant and underestimates the bulk modulus. Then only the LSDA+ U method accurately reports the

electronic and structural properties of NiO. The calculated band gap and the density of states (DOS)

show that the material NiO is the charge-transfer insulator, which agrees with the spectroscopy data.

The comparison between the charge density of LSDA (not considering U) and that of LSDA+U

(considering U) demonstrates that the trend of ionic crystal for NiO is obvious.

Crown Copyright & 2008 Published by Elsevier B.V. All rights reserved.

1. Introduction

For many materials, the density-functional theory (DFT) [1]

within the local-spin-density approximation (LSDA) [2] provides a

good description of their ground-state properties. However, the

problems arise when the DFTLSDA approach is applied to

materials with the ions that contain incomplete d or f shells,

such as transition-metal oxides or heavy fermion systems. For

example, the standard LSDA approximation applied to NiO allows

the appearance of a forbidden band, which explains the insulating

property, but the forbidden gap values are very smaller than the

experimental estimations [3]. Furthermore, in the MottHubbard

picture of NiO, the dd Coulomb interaction splits the Ni d sub-bands into the so-called lower and upper Hubbard bands. The

upper Hubbard band has mostly Ni 3d9 character, while the top of

the valence band has a 3d8 character, leading to a MottHubbard

gap of dd type. However, in contrast to the MottHubbard model,

the energy band gap caused by the Ni 3d correlation is the charge-

transfer type between the occupied oxygen 2p and the Ni 3d

empty states, which is approved by X-ray absorption [4], X-ray

photoemission and bremsstrahlung isochromat spectroscopies

data [5] on LixNi1xO. Due to the failure within standard LSDA

approximation, some improved methods have been created by a

few physical scientists.

Several attempts have been proposed to improve the LSDA

approach among which the generalized gradient approximation

(GGA), the self-interaction correction (SIC), the HartreeFock (HF)

method, the GW approximation (GWA), the orbital polarized

correction and the LSDA are corrected by the on-site Coulomb

interaction U(LSDA+U). The GGA [6], which takes into account the

radial gradient corrections, can only open a small band gap [7].

Then the SIC [810] eliminates the spurious interaction of an

electron with itself from the conventional DFTLSDA method.

Compared to the LSDA, the band gap and the magnetic moments

are significantly increased. However, the band gap still is toosmall, and the SICLSDA method predicts a larger energy band gap

for NiO than for FeO and CoO, in contrast to the experimental

result [11]. The HF method [12] is appropriate for describing Mott

insulators. However, a serious problem of the HF approximation is

the unscreened nature of the Coulomb interaction. The so-called

value of Coulomb interaction parameter U is rather large

(1520 eV) while screening in a solid leads to much smaller

values: 8 eV or less [13,14]. Due to the negligence of screening, the

HF energy gap values are 23 times larger than the experimental

values [12]. For the GWA [15], the problem of screening is

addressed in a rigorous way, which may be regarded as an HF

theory with a frequency- and orbital-dependent screened Cou-

lomb interaction. The GWA has been applied with success to real

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journal homepage: www.elsevier.com/locate/physb

Physica B

0921-4526/$ - see front matter Crown Copyright& 2008 Published by Elsevier B.V. All rights reserved.doi:10.1016/j.physb.2008.10.009

Corresponding author. Tel./fax.: +86 28 85405234.

E-mail address: [email protected] (T. Gao).

Physica B 404 (2009) 8994
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reproduce experimental results and determining which type of

LSDA (GGA)+U is better able to provide an accurate description

of NiO.

3.1. Fixing the U

For NiO, the values of U and J, which have been calculated by

the different methods, are listed in Table 1. The LAPW methodcalculation achieved by Madsen and Novak [25] evaluates U to

7.3 eV, and Anisimov et al. [17] performing an LMTOASA method

ascribes 8 eV to U. However, the value 5.4 eV of U is estimated

within the LCAO method by Hugel and Kamal [19]. Our estimation

for Uis 6.2 eV based on the FLAPW method. But the experimental

value ofUis 5.8 eV from the spectroscopic data [30]. It is clear that

our estimation for Uis in better agreement with the experimental

value than others.

3.2. Geometry, elastic and magnetic properties

The optimized lattice constant a (A), the calculated spin

magnetic moments ms (mB), the bulk modulus B (GPa) and the

corresponding experimental data are listed in Table 2. The resultsdemonstrate that the LSDA+U provides a better estimate of the

lattice constant than LSDA compared with the experiment, the

lattice constant is more severely overestimated by GGA(96)+U

than by GGA(96), and the GGA(06)+U also overestimates the

lattice constant. It is clear that the optimized lattice constant

(GGA96+U) 4.30A is the biggest while the (LSDA) 4.075 A is the

smallest in all the optimized lattice constants, which is consistent

with the result that the LSDA systematically underestimates the

equilibrium lattice constants [31]. In terms of the spin magnetic

moments, the LSDA+U, GGA(96)+U and GGA(06)+U provide good

results, which are in agreement with the biggest experiment and

overestimate ms by 0.1, 0.09 and 0.1mB, respectively. For the bulk

moduli, the LSDA+U provides an improved calculated result

compared with LSDA, although both results overestimate B. We

cannot confirm which type can accurately estimate B due to too

many different experimental values. However, as a whole, the

LSDA+U can more accurately describe the structural propertiesand bulk moduli than the other methods.

3.3. Density of states

The calculated densities of states (DOS) and the projected

densities of states (PDOS) of Ni 3d states and O 2p states are

shown in Fig. 2 (LSDA+U), Fig. 3 (LSDA), Fig. 4 (GGA96+U), Fig. 5

(GGA96), Fig. 6 (GGA06+U) and Fig. 7 (GGA06). Figs. 3, 5 and 7,

show that the ground state of NiO is regarded as the metal, which

is unacceptable for the DFTLSDA (GGA96, GGA06). However,

Fig. 2 (LSDA+U), Fig. 4 (GGA96+U) and Fig. 6 (GGA06+U), can

explain that the ground state of NiO is the insulator. The band

gaps, which are calculated from Figs. 2, 4 and 6, mainly between O

2p states at the top of the valence band and primarily Ni 3d statesat the bottom of the conduction band are listed in Table 3. For NiO,

the ground state is the charge-transfer insulator, which is seen

from the band gap in Figs. 2, 4 and 6, which was interpreted by the

ARTICLE IN PRESS

Table 1

Values of U (eV) and J (eV) calculated with the different methods.

U J

FLAPW (present) 6.2 1.36

LAPWa 7.3 1.36

LMTOASAb 8.0 0.95

LCAOc 5.4 1.12

Expt.d 5.8

a Madsen and Novak,[25].b Anisimov et al. [17].c Hugel and Kamal [19].d Fuggle et al. [30].

Table 2

Calculated and experimental values of the optimized lattice constant a (A),

calculated spin magnetic moments ms (mB) and bulk moduli B (GPa).

a (A) ms (mB) B (GPa)

LSDA 4.075 0.05 296.2

LSDA+U 4.180 2.00 202.2

GGA(96) 4.236 1.90 169.9

GGA(96)+U 4.300 1.99 167.9

GGA(06) 4.158 1.65 196.1

GGA(06)+U 4.236 2.00 202.9

Expt. 4.168a 1.64b, 1.77c, 1.90d 145e, 190f, 199g, 205h

a Wyckoff[32].b Alperin [33].c Fender et al. [34].d Barin [35].e Duplessis [36].f Notis [37].g Clenenden and Drickamer [38].h Wang [39].

4

2

0

-2

-4

-60.9

0.60.30.0

-1

-232

1

0

-2

-4

-10 -5 0 5 10 15

DOS/(1/eV)

Energy (eV)

EF Total

O-2p

Ni-3d

Fig. 2. The density of states of NiO calculated with LSDA+U.

8642

0-2-4

-6

1.2

0.80.40.0

-0.4

-0.8

-1.26

4

2

0

-2-4

-6

-10 -5 0 5 10 15

DOS

(1/eV)

Energy (eV)

EF Total

O-2p

Ni-3d

Fig. 3. The density of states of NiO calculated with LSDA.

T. Cai et al. / Physica B 404 (2009) 8994 91


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spectroscopy data [40]. The Dn (the number spin upthe number

spin down), which is calculated from the DOS in Figs. 2, 4 and 6,

are 1.98, 2.00 and 2.01, respectively. It is found that Dn is in good

agreement with the corresponding calculated spin magneticmoments data.

The calculated and experimental band gap within the different

basis sets and methods is shown in Table 3. The calculated results

illuminate that our calculating band gap, 4.04 eV (LSDA+U),

4.31 eV (GGA96+U) is in better agreement with the experimental

values than the results of the LMTOASA [41] and FLMTO [42]

methods from Table 3. The band gap 3.4 eV (GGA06+U) as well as

the 3.1 eV (LMTOASA, LDA+U) and 3.4 eV (FLMTO, LDA+U) are

smaller than the experimental data. The calculated data 4.04 eV

(LSDA+U) are approximately equal to the data 4.1 eV (LCAO,

LSDA+U) [19]. Taken as a whole, the results show that the LSDA+U,

GGA96+U and GGA06+U all provide a good description of the

ground state for NiO; however, the band gap data indicate that

superior result is obtained with the LSDA+U method.

3.4. Charge distributions and electron localization

For the (0 0 1) plane of the NiO crystal structure, the charge

density distributions are plotted in Fig. 8 (LSDA), Fig. 9 (LSDA+U),

Fig. 10 (GGA96), Fig. 11 (GGA96+U), Fig. 12 (GGA06) and Fig. 13

(GGA06+U). It is clearly shown that the value of the isosurface is

marked between the Ni atoms and the O atoms in each figure.

Compared to Fig. 8 (LSDA), the corresponding value of the

isosurface between the Ni atoms and the O atoms decreases in

Fig. 9 (LSDA+U), which illustrates the trend of the ionic crystal for

material NiO. Obviously, the same result is found in Fig. 10

(GGA96) and Fig. 11 (GGA96+U), while the same result is not seen

in Fig. 12 (GGA06) and Fig. 13 (GGA06+U). The reason may be that

the correction of U is too little for the GGA06. Finally, Fig. 11

(GGA96+U) could powerfully describe the electron localization in

which the value of the isosurface between the Ni atoms and the Oatoms is least in all the figures. The above results demonstrate

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4

2

0

-2

-4

0.9

0.6

0.3

0.0

-1.6

-0.8

3

2

1

0

-2

-4

-10 -5 0 5 10 15

DOS(

1/eV)

Energy (eV)

EF Total

O-2p

Ni-3d

Fig. 4. The density of states of NiO calculated with GGA96+ U.

6

4

20

-2

-4

-60.9

0.6

0.00.3

-0.3-0.6-0.9-1.2-1.5

4

2

0

-2

-4

-10 -5 0 5 10 15

DOS

(1/eV)

Energy (eV)

EF Total

O-2p

Ni-3d

Fig. 5. The density of states of NiO calculated with GGA96.

4

2

0

-2

-4

-60.90.60.30.0

-0.5

-1.0-1.5

3

2

1

0

-2

-4

-10 -5 0 5 10 15

DOS

(1/eV)

Energy (eV)

EF Total

O-2p

Ni-3d

Fig. 6. The density of states of NiO calculated with GGA06+U.

6420

-2

-4

-6

0.91.2

0.6

0.0

0.3

-0.5

-1.0-1.5

5.0

2.5

0.0

-2

-6

-4

-10 -5 0 5 10 15

DOS(

1/eV)

Energy (eV)

EF Total

O-2p

Ni-3d

Fig. 7. The density of states of NiO calculated with GGA06.

Table 3

Calculated and experimental band gap (Eg in eV).

Basis

set

FLAPW LMTOASAa FLMTOb LCAOc

Method LSDA+U GGA(96)+U GGA(06)+U LDA+U LDA+U LSDA+U Expt.

Eg 4.04 4.31 3.4 3.1 3.4 4.1 4.0d,4.3e

a Anisimov [41].b Dudarev [42].c Hugel and Kamal [19].d Hufner et al. [43].e Sawatzky and Allen [44].

T. Cai et al. / Physica B 404 (2009) 899492


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that the LSDA+U and GGA96+Ucan better describe the ionicity of

NiO than the LSDA and GGA96, and the GGA96+Uprovides a littlebetter description for ionicity than the LSDA+U method.

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Fig. 8. The charge density in the (0 0 1) plane of NiO with LSDA.

Fig. 9. The charge density in the (0 0 1) plane of NiO with LSDA+U.

Fig. 10. The charge density in the (0 0 1) plane of NiO with GGA96.

Fig. 11. The charge density in the (0 0 1) plane of NiO with GGA96+U.

Fig. 12. The charge density in the (0 0 1) plane of NiO with GGA06.

Fig. 13. The charge density in the (0 0 1) plane of NiO with GGA06+U.

T. Cai et al. / Physica B 404 (2009) 8994 93


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4. Conclusions

The first-principles all-electron FLAPW technique has been

used to perform the high-precision calculations of NiO within the

LSDA (GGA) and LSDA (GGA)+U methods, respectively. The

estimated value U (6.2eV) based on the FLAPW method is well

in agreement with the experimental data (5.8 eV). The ground-

state properties of NiO were calculated using the LSDA (GGA) andLSDA (GGA)+U (6.2 eV) methods.

The above results shed light on how the LSDS (GGA)+U

calculations affect various ground state properties of NiO. In

short, the optimized lattice constant (4.180 A), calculated spin

magnetic moments ms (2.00mB), bulk moduli B (202.2 GPa) and

band gap (4.04 eV) by the LSDA+Umethod are in better agreement

with the corresponding experimental data than the other

methods. Thereby, we will investigate whether the FLAPW and

LSDA +Umethods are applicable to all the transition-metal oxides,

and then the results of those calculations will be reported in

future publications.

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