ARTICLES

Sci. Rep. Kanazawa Univ.
Vol. 64, pp. 1–13, 2020

Electronic structure investigation of spinel NiCo2O4
from quasi-particle self-consistent GW method
Hasan Al Rasyid*,,1,2 Masao Obata,1,3 Indra Pardede,3,4 Marleni Wirmas,1Takao Kotani,5 Tatsuki Oda1,3
1 Graduate School of Natural Science and Technology, Kanazawa University

Kanazawa, 920-1192, Japan
2 Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung

Bandung, 40132, Indonesia
3 Institute of Science and Engineering, Kanazawa University

Kanazawa, 920-1192, Japan
4 Department of Physics, Institut Teknologi Sumatera (ITERA)

Lampung Selatan, 35365, Indonesia
5 Department of Applied Mathematics and Physics, Tottori University

Tottori, 680-8552, Japan(Receieved June 19, 2020 and accepted in revised form August 12, 2020)

Abstract Quasi-particle self-consistent GW (QSGW) electronic structure calculation was
performed in three spinel structures of ternary magnetic compound NiCo2 O4 , i.e. normal
spinel and two of inverse spinel structures (Types A and B). The QSGW method is stateof-the-art on electronic structure investigation without using any empirical parameter. The
results were also compared with those from the density functional theory (DFT) based on
generalized gradient approximation (GGA). Half-metallicity was observed in all cases investigated, where the band energy gap appears in the majority spin state. The QSGW indicatedthat the gap is larger than that of the corresponding GGA case. The QSGW may yield more
localized 𝑑-orbitals, compared with those from the GGA. Based on the projected density ofstates (PDOS) analysis, Co atoms of the octahedral site [Co(oct)] were found to be a nonmagnetic/weak magnetic conﬁguration (low spin state) among them, resulting in a negligible
contribution to the total magnetization. The largest contribution to the total magnetization isprovided by Co atoms of tetrahedral [Co(tet)] site (high spin state). The QSGW indicatedthe diﬀerence of octahedral crystal ﬁelds on the diﬀerent cation sites of Co(oct) and Ni(oct).
The electronic states near the Fermi level consist mainly of Co(tet) 3d 𝑒 g -orbitals. In thenormal and inverse (type A) spinels, the Fermi level crosses the states with the DOS having
a large sharp peak, implying instability of the electronic structure. The results of QSGW
were discussed by comparing with those of one-shot GW and other methods.
Keywords.
cobaltite.

QSGW method, DFT method, ab-initio electronic structure, spinel, nickel* Corresponding author

Email: hasanalrasyid.har@gmail.com

2

1

Electronic structure of spinel NiCo2 O4 : QSGW method

Introduction

Oxygen reduction reaction (ORR) is one of the most basic reactions that construct many complex
reactions such as energy generation (fuel cell), weathering of materials, and most of the biologicalprocesses [1]. One of the main interests of this reaction focused on its role in the oxygen electrode used in electrical-power related systems, i.e. metal-air batteries and fuel cells. From thispoint of view, being an abundantly available transitional metal oxide, Nickel cobaltite, NiCo2 O4 ,shows many advantages, compared to already available alternatives. In catalysis, NiCo2 O4 has arole as a heterogeneous catalyst with the reaction of ORR. However, the mechanism of ORR on
metal surfaces remains unclear. NiCo2 O4 indicates a half-metallic property, according to low ﬁeld
magnetoresistance (LFMR) measurements [2]. Nickel cobaltite is also reported to be half-metallic
with −19.1% low-ﬁeld magnetoresistance at 0.5 T and −50% at 9 T (both measured at 2 K) [2].
Combined with its high Curie temperature (295K [3], 350K [4]), these properties made nickel
cobaltite into a good candidate as spintronic materials. One can easily tune its electrical and magnetic properties by varying the crystal growth temperature and oxygen pressure [5]. The interest
on NiCo2 O4 mainly comes from its abundance and its potential use, as spintronics application, as
well as an alternative electrode used in electrical-power related systems, i.e. metal-air batteries
and fuel cells. NiCo2 O4 is one of the promising materials to be used in applications of catalysis
and spintronics.
NiCo2 O4 is a ferrimagnet [4, 6] and one candidate of novel materials that could be engineeredin many ways to exhibit diﬀerent properties. By utilizing temperature growth of less than 450◦ C,
one can acquire metallic-ferrimagnetic NiCo2 O4 thin ﬁlm. On the contrary, with more than 450◦ C,
we would acquire thin ﬁlm as a non-magnetic insulator [7]. This uniqueness is believed to comefrom the competition between double exchange interaction among cations with diﬀerent charges
and superexchange interaction among those with the same charges [3].
An electronic structure investigation from a theoretical approach was reported in the work
combined with the experimental measurement [8]. The authors have used a modiﬁed BeckeJonson formula in the exchange-correlation function based on local spin density approximation(mBJ-LDA). It reported large energy gaps of around 3−4 eV in majority spin states between thevalence and conduction bands, and the result indicated a half-metallic property. The properties
of electronic structure, such as large energy gap, have not been validated yet and the details on
the electronic structure are not so clear that they are self-consistently understood with the properties of catalysis. The successive theoretical approach, namely the LDA+U approach, introduced
an on-site Coulomb parameter of the Hubbard model, 𝑈eﬀ , to the cation sites [2, 5]. The resultsof the spin-polarized density of states indicated a half-metallic property with an energy gap in
the majority-spin state of about 1.3 eV. The analysis of atomic and orbital projections revealed
electrons of the Ni(Oct) and Co(Tet) orbitals hybridized with O at the Fermi level. The obtained
pictures on electronic and magnetic structures at the Fermi level may be based on the understanding or interpretation of many complicated experimental observations [2, 5]. This means that one
can get some satisfaction from such a theoretical approach to its half-metalicity. However, in those
approaches, the empirical parameters were used for the 𝑈eﬀ on Ni and Co sites which were introduced as Fe spinel oxides [9]. The semiconducting/insulating property of NiCo2 O4 has not been
discussed from theoretical electronic structure results.
We have theoretically investigated the electronic and magnetic properties of bulk NiCo2 O4 tounderstand its basic features. In our theoretical investigation, quasi-particle self-consistent GW(QSGW) calculations [10, 11, 12] were performed in normal and inverse spinels of NiCo2 O4 .
This method does not employ any empirical parameters such as 𝑈eﬀ . This feature indicated a lot

Hasan Al Rasyid, Masao Obata, Indra Pardede, Marleni Wirmas, Takao Kotani, Tatsuki Oda

3

Table 1: Conﬁgurations of tetragonal and octahedral sites for X and Y elements in the spinel
structure XY2 O4 . The numbers in parentheses are the number of atoms per formula unit.
systemNormalInverse type A
Inverse type B

tetrahedral site (2)X(2)X(1),Y(1)Y(2)

octahedral site (4)Y(4)X(1), Y(3)X(2), Y(2)of advancements in the electronic structure analyses of semiconducting and insulating materials.
The QSGW method enables the calculation of metallic electronic structures. Such an advantage
may be useful for a wide range of materials in the advanced methods beyond the density functional
theory (DFT) [13] approach. The performance of QSGW beyond DFT in improving the electronic
structure of semiconductors has been well known already [11]. The recent progress has reached toa group of materials for power electronics [14, 15] and a group of surfaces [16]. We can ﬁnd moredata about applicability on its performance on wider sets of materials, i.e. half-metallic, magnetic,
and metallic materials.
The half-metallic property was found on all the results of NiCo2 O4 and the metallicity is associated mainly with the 3d orbitals on tetrahedral Co(tet) in the minority spin state. The QSGW
investigation revealed that the Ni 3d components at the octahedral site are largely suppressed at
the Fermi level by a mixed-valence feature with the hybridization with O 2p orbital.

2

Crystal structure and computation method

2.1

Normal and inverse spinel structures

Spinel materials or their family are usually presented as a chemical formula XY2 O4 , where X andY are cationic elements. For the cationic elements, there are two distinguishable atomic sites. They
have tetrahedral and octahedral symmetries, called tetrahedral and octahedral sites, respectively.
There are 2 tetrahedral sites and 4 octahedral sites in a primitive unit cell (two formula units).
In the normal spinel, all of X occupies the tetragonal site and all of Y at the octahedral site. In
inverse spinel structure, X and Y atoms are interchanged with each other between the tetrahedral
and octahedral sites. As in the previous work of NiCo2 O4 [7], we considered two types of inverse
spinel, called Types A and B (inverse A and inverse B). The conﬁgurations of tetrahedral andoctahedral sites for X and Y elements are presented in Table 1.
The crystal structure of the normal spinel belongs to a faced-centered cubic structure with thespace group of No. 227. In the case of NiCo2 O4 , its structure was reported to have an inverse
spinel structure [6]. We calculated normal and two inverse spinel structures of NiCo2 O4 . The
normal and inverse spinel structures (inverse A and inverse B) are presented in Figure 1. Silwal
et al. [7] suggested that the inverse B conﬁguration has a higher tendency to happen rather thaninverse A conﬁguration. Our discussion in this paper particularly focused on the inverse type B.
Figure 2 picks up atomic conﬁgurations at the tetrahedral and octahedral sites in the spinel
structure. Each cation atom is surrounded by four or eight oxygen atoms depending on these sites.
Oxygen imposes a strong crystal ﬁeld at the cation site. As a consequence of such crystal ﬁeld,
the ﬁve energy levels of 3d orbital will split into two kinds of levels denoted as 𝑡2g (3 levels) and𝑒 g (two levels). For the tetrahedral site surrounded by negatively charged oxygen, the 𝑡2g levels arelowered and those of 𝑒 g are raised. In the octahedral site, this level arrangement is reversed. The

4Electronic structure of spinel NiCo2 O4 : QSGW method

Figure 1: Three spinel structures of NiCo2 O4 . From left to right, they indicate spinels of normal,
inverse type A, and inverse type B. Ni, Co, and O atoms are represented by silver, blue, and redcolor respectively.

Figure 2: Atomic conﬁgurations of tetrahedral (left) and octahedral (right) sites, respectively.
Silver or blue balls denote a cation atom and red balls denote oxygen atom.

energy width of this splitting may indicate the degree of the crystal ﬁeld.
We investigated the optimized structure of the lattice parameter 𝑎 and internal parameter 𝑢using the house-package of DFT(GGA) code [17] for the normal spinel structure of NiCo2 O4 . As
𝑢 increases, the oxygen tetrahedron around the cation of the tetrahedral site expands, resulting in an
elongation of the cation-oxygen bond. They were determined to be 𝑎 = 8.115 Å and 𝑢 = 0.3881.
The former was in agreement with the experimental values. Thus, we decided to use the set ofexperimental values in our further research. Indeed, we took the experimental values of 𝑎 = 8.114
Å and 𝑢 = 0.3833 [6, 18]. The diﬀerence (𝛿𝑢) between two√ internal parameters presented above
corresponds to an oxygen atomic distortion of 0.07 Å ( = 3𝑎 𝛿𝑢). This is much smaller than astandard bond length of Ni-oxide (0.4%) [6].

2.2

Calculation procedure

In the QSGW calculation, the energy level (𝜀 𝑖 and |Ψ𝑖 ⟩: eigenvalue and eigenstate) of quasiparticle (QP) is obtained by solving the equation
]
[ 2
ℏ 2
(2.1)
−
∇ + 𝑉ext + 𝑉H + 𝑉xc |Ψ𝑖 ⟩ = 𝜀 𝑖 |Ψ𝑖 ⟩,2𝑚

Hasan Al Rasyid, Masao Obata, Indra Pardede, Marleni Wirmas, Takao Kotani, Tatsuki Oda

5where 𝑉ext , 𝑉H , and 𝑉xc are the potentials from nuclei, the electrostatic potential from electrons, andexchange-correlation potential, respectively. 𝑉xc is self-consistently evaluated from the following
formula:
[
]
1∑
𝑉xc =
|Ψ𝑖 ⟩ Re{Σ(𝜀 𝑖 ) + Σ(𝜀 𝑗 )} ⟨Ψ 𝑗 |,(2.2)
2 𝑖, 𝑗where Re represents the Hermitian part and Σ(𝜀 𝑖 ) is derived from the 𝐺𝑊 self-energy (Σ ≡ 𝑖𝐺𝑊)
in 𝐺𝑊 approximation (GWA). In the practical evaluation, the screened Coulomb interaction 𝑊 isevaluated through the polarization function 𝑃 within random phase approximation. Once the new
set of 𝜀 𝑖 and |Ψ𝑖 ⟩ is obtained, it is used in the evaluation of 𝑃 and 𝑊 for the next iterative cycle.
The details of GWA are referred to the literature [10, 11, 12].
For the QSGW calculation, we used the package ecalj developed by Kotani et al. [19, 20].
𝑃 in the QSGW calculation was performed using a 4×4×4 k-mesh sampling. The self-consistentcalculation of QSGW was performed by an iterative procedure with a starting data constructedfrom DFT result. The iteration of QSGW was continued until a convergence on the maximum
change in eigenvalue 𝜀𝑖 within 0.01 eV. Such convergence was achieved with around 10 iterations.
In this work, we discussed the electronic structures based on the results of GGA, one-shot GW, and
QSGW, where the one-shot GW is assigned to the ﬁrst step of QSGW iterative cycles. We analyzed
band dispersion curves, total and projected densities of states, and magnetic conﬁgurations.
The post-processing was done using in-house scripts with the help of parallelization implemented by GNU Parallel [21]. VESTA software package was employed to generate all crystal
structure visualization [22]. The Fermi surface was produced using the XCrysDen software package [23].

3Results and discussions

3.1

Electronic structure

The energy band dispersions of inverse B are shown in Figure 3. These ﬁgures are results for
GGA, one-shot GW, and QSGW methods. In general, there are several features revealed by theseenergy bands. All calculations revealed half-metallic properties of NiCo2 O4 , where an energy gap
opens at the Fermi energy in the majority spin state. For the minority spin state two bands acrossto the Fermi level in QSGW, whileas three bands in GGA and one-shot GW.
As shown in Figure 3, the valence band around the Fermi level consists of the three parts ofoxygen orbital and two cation 3d orbitals. The oxygen part is separated from the cation parts withan energy gap, and in the majority spin state of GGA the cation parts are further separated to theoccupied and unoccupied states below and above the Fermi level, respectively. In the one-shot
GW, the cation part is separated into three parts in both of the majority and minority spin states.
However, during the self-consistent calculation of QSGW, the cation part once separated at the
one-shot GW is merged partially. As a whole property, the bandwidth of cation bands becomesnarrow from GGA to QSGW.
Total and projected density of states (PDOS) obtained from GGA, one-shot GW, and QSGW
are shown in Figure 4. All of PDOS display metallic behavior only at the minority spin state.
The cation band around the Fermi level is distributed supposing the crystal ﬁeld of the octahedralor tetrahedral site. Such a crystal ﬁeld induces the energy splitting between 𝑡2g and 𝑒 g states.
The DOS shape of the cation part becomes sharper from the result of GGA to those of one-shot
GW and QSGW. In these GW results, the crystal ﬁeld eﬀect appears more clearly than those of

6Electronic structure of spinel NiCo2 O4 : QSGW method

Energy−EF (eV)GGA

One Shot QSGW

QSGW

2

2

2

0

0

0

−2

−2

−2

−4

−4

−4

−6

−6

−6

−8

−8

−8

Γ

L U X W L1 Γ
Wave Vector

X

U1

Γ

L U X W L1 Γ
Wave Vector

X

U1

Γ

L U X W L1 Γ
Wave Vector

X

U1

Figure 3: The energy band dispersions for NiCo2 O4 with the inverse spinel type-B structure,green indicates minority spin states and magenta for majority ones. From left to right, the resultsof GGA, one-shot GW, and QSGW. The Fermi level presented by the horizontal line is set at zeroenergy.

GGA. This strongly indicates that the Coulomb interaction aﬀected by O is taken into accountmore accurately through the screened Coulomb interaction 𝑊. Such eﬀect may vary dependingon the symmetry (octahedral or tetrahedral) of cation site. Particularly in the majority spin state,
the conduction band above 𝐸 F in QSGW reveals a band splitting between Ni 3d and Co 3d atoctahedral sites. This split may be attributed to the diﬀerence in octahedral crystal ﬁelds. The
splitting of 𝑡2g − 𝑒 g on Ni is much larger than that on Co(oct). Our results imply that the QSGW
method can provide distinct crystal ﬁelds for diﬀerent cations. For the tetrahedral site [Co(tet)],eﬀective crystal ﬁeld splitting seems to be small in the results of GGA, one-shot GW, and QSGW.
The sharp PDOS of cation bands may be due to the localization of electron wave functions
for the transition metals. This feature is a typical consequence resulting from the GWA. Anotherconsequence appearing in Figure 4 is observed in the oxygen band. That band is shifted to lower
energy, implying a stabilization of oxygen band in GWA.
The half-metallicity in the present work is in agreement with the previous theoretical results
including the mBJ-LDA [8] and the LDA+U approaches [2, 5]. Overall DOS from the QSGW
is similar to those from the mBJ-LDA and LDA+U, whereas, in the later, the oxygen band is
not separated from the cation band. In regards to the cation band, the upper cation bands are
shifted to the higher energies compared with the previous works mentioned above. The DOS ofminority spin in the vicinity of the Fermi level is overall similar to those in the previous works but
is diﬀerent from them in quantitative detail.
We note some features during the self-consistent iterative cycle by comparing the PDOS between one-shot GW and QSGW. First, the degree of energy splitting from the crystal ﬁeld increases
in the self-consistent process. Second, as a result, the sharpness of DOS in one-shot GW becomes
broad, but have never reached to the broadness of DOS-like GGA.
Figure 5 reported the total DOS and PDOS of normal spinel and inverse spinel (type A). As
described before, the half-metallic structure is observed. The energy level splitting caused by the
crystal ﬁeld is observed like those of the inverse spinel (type B), while the Ni(tet) component is7Hasan Al Rasyid, Masao Obata, Indra Pardede, Marleni Wirmas, Takao Kotani, Tatsuki Oda

Co 3d

15
10
5
0
-5
-10
-15

CoTd 3d

15
10
5
0
-5
-10
-15

Density of States (states/eV/cell)

Density of States (states/eV/cell)

Ni 3d

15
10
5
0
-5
-10
-15

O 2p

15
10
5
0
-5
-10
-15
-8

-6

-4

-2

0

Total

15
10
5
0
-5
-10
-15

Ni 3d

15
10
5
0
-5
-10
-15

Co 3d

15
10
5
0
-5
-10
-15

CoTd 3d

15
10
5
0
-5
-10
-15

O 2p

15
10
5
0
-5
-10
-15
2

Energy-EF (eV)

-8

-6

-4

-2

Energy-EF (eV)

0

Total

15
10
5
0
-5
-10
-15

Density of States (states/eV/cell)

Total

15
10
5
0
-5
-10
-15

Ni 3d

15
10
5
0
-5
-10
-15

Co 3d

15
10
5
0
-5
-10
-15

CoTd 3d

15
10
5
0
-5
-10
-15

O 2p

15
10
5
0
-5
-10
-15
2

-8

-6

-4

-2

0

2

Energy-EF (eV)Figure 4: Total and projected densities of states (DOS) of the Ni(oct) 3d, Co(oct) 3d, Co(tet) 3d,
and O 2p orbitals of inverse type B spinel NiCo2 O4 for GGA, one-shot GW, and QSGW.
missing in the latter case. An important diﬀerence against the inverse type B is behavior at the
Fermi level, as shown in Figure 6. Unlike the case of inverse type B, the Fermi level crosses alarge peak of DOS. These behaviors strongly imply that such a metallic system has the possibility
to take some form of instability. In other words, the electronic state can be stabilized further bysome induction like a picture of the Jahn-Teller eﬀect. In the view of instability, the normal spinel
or inverse type A may be at a higher energy state compared with the inverse type B.

3.2

Magnetic conﬁguration

All of the methods (GGA, one-shot GW, and QSGW) provide a net magnetization of 2 𝜇B /f.u.
(f.u.: formula unit.) in the inverse spinel structures of type A and type B, and also normal spinel
structure. This value of net magnetization has been reported in the previous work [4]. Both inverse
structures provide a ferrimagnetic arrangement, whereas the normal spinel structure exhibits ferromagnetic arrangement. However, the ferrimagnetic arrangement provided by the inverse type A
has a signiﬁcant diﬀerence with that of type B. This is due to the diﬀerence of cation conﬁguration
between the inverse type A and type B.
Our total magnetization is much larger than the experimental value (1.25 𝜇B /f.u.) [6]. Recent
experiment [2] reported that the total magnetization gradually increases to a larger value (1.84

8Electronic structure of spinel NiCo2 O4 : QSGW method
Total
15

Total

15
10
5
0
-5
-10
-15

10
5
0
-5
-10

Ni 3d
15

5

-5
-10
-15
Co 3d
15
10
5
0
-5
-10

O 2p
15
10
5
0
-5
-10
-15
-6

-4

-2

0

CoOct 3d

15
10
5
0
-5
-10
-15

NiTd 3d

15
10
5
0
-5
-10
-15

CoTd 3d

15
10
5
0
-5
-10
-15

-15

-8

Density of States (states/eV/cell)

Density of States (states/eV/cell)

100

NiOct 3d

15
10
5
0
-5
-10
-15

-15

2

O 2p

15
10
5
0
-5
-10
-15
-8

Energy-EF (eV)

-6

-4

-2

0

2

Energy-EF (eV)Figure 5: The total and projected densities of states of the Ni 3d, Co 3d, and O 2p orbitals of
normal spinel (left) and inverse type A spinel (right) of NiCo2 O4 from QSGW.
𝜇B /f.u.) as the annealing time of specimen increases. This diﬀerence in the magnetization may be
attributed to a disordered crystal structure in the experimental specimen.
The magnetic moment on each atom varied from GGA to QSGW, as shown in Table 2. The
ferrimagnetic conﬁguration of inverse Type B is clearly indicated. The QSGW consistently showsa larger absolute value for each atomic magnetic moment except for Co(oct). This behavior can beunderstood since the QSGW tends to indicate a more localized electronic state in 𝑑-orbital states.
In the results of QSGW, the atomic magnetic moments of inverse B are 2.77 𝜇B for Co(tet) and−1.39 𝜇B for Ni. These results are comparable with the experimental data of Co(tet) 2.18 andNi −1.49 𝜇B [24] and the theoretical LDA+U data of 2.39 𝜇B for Co(tet) and −1.12 𝜇B for Ni[25] with 𝑈eﬀ [Ni, Co(tet) and Co(oct): 6.6, 4.7, and 6.7 eV, respectively, determined from a ﬁrstprinciples linear response theory]. One of the interesting results is the induction of non-negligible
magnetic moment on O 2p orbitals. This contribution is estimated to be 0.40 𝜇B in the formulaunit, resulting from the hybridization between cation 3d and O 2p orbitals. The occurrence ofmagnetic moment on O atoms implies the importance of its roles for understanding the electronic
structure in NiCo2 O4 . The moment of Co(oct) indicates a small value, which corresponds to a low
spin state of Co atom.
In Table 3, magnetic moments of normal and inverse type A are reported. The ferromagnetic

Hasan Al Rasyid, Masao Obata, Indra Pardede, Marleni Wirmas, Takao Kotani, Tatsuki Oda
NiOct t2g

15
10
5
0
-5
-10
-15

CoOct eg

15
10
5
0
-5
-10
-15

CoTd t2g

15
10
5
0
-5
-10
-15

CoTd eg

15
10
5
0
-5
-10
-15
-8

-6

-4

-2

Energy-EF (eV)02

NiOct eg

15
10
5
0
-5
-10
-15

Density of States (states/eV/cell)

Density of States (states/eV/cell)

CoOct t2g

15
10
5
0
-5
-10
-15

NiOct t2g

15
10
5
0
-5
-10
-15

NiOct eg

15
10
5
0
-5
-10
-15

9

CoOct t2g

15
10
5
0
-5
-10
-15

CoOct eg

15
10
5
0
-5
-10
-15

CoTd t2g

15
10
5
0
-5
-10
-15

CoTd eg

15
10
5
0
-5
-10
-15
-8

-6

-4

-2

0

2

Energy-EF (eV)Figure 6: 𝑡2g and 𝑒 g components for the projected density of states on Ni 3d and Co 3d orbitals ininverse type B spinel of NiCo2 O4 from GGA and QSGW.
and ferrimagnetic conﬁgurations are indicated respectively. The moment of Ni(tet) is coupledferromagnetically with those of Co(tet), while that of Ni(oct) is coupled antiferromagnetically.
In all of the spinels investigated, the tetrahedral cation always contributes to the ferromagnetic
component, while the octahedral cation contributes to the nonmagnetic or antiferromagnetic component. Additionally, the oxygen, even though the degree is small, contributes to the ferromagnetic
component. Interestingly, the sum of atomic components is commonly similar among the spinel
structures investigated.

3.3

Electron conﬁguration

The atomic magnetic moment is the consequence of electron conﬁguration at its atomic site. In
the analysis of PDOS, the 3d electron conﬁguration is tabulated for the inverse B in Table 4. These
numbers of electrons are estimated within each atomic region. Therefore, the electron outside theatomic region is excluded from the value. On the comparison between the results of GGA and
QSGW, several features are pointed out. First, the number is decreased at cation 3d sites and O 2p
sites. Second, the number of atomic majority (minority) spin state increases (decreases) at CoTd
3d and Ni 3d. Both features result in the increase of magnetic moment on CoTd 3d and Ni 3d.
The electron conﬁguration of the atom in the solid crystal may not be the best way to represent10Electronic structure of spinel NiCo2 O4 : QSGW method

using the ionic representation for cation elements. However, such a way is helpful to understandpictures of electronic state more deeply. As speculation, the electron conﬁguration at the atomic
state may be presented as Co(Td) 3d (d↑)5 (d↓)2 , Ni 3d (d↑)3.5 (d↓)5 , and Co 3d (d↑)3.5 (d↓)3.5 .

3.4

Properties at the Fermi level

The electronic states near the Fermi level consist mainly of the 3d-orbitals on Ni(oct) and Co(oct)in the majority spin state, and of the 3d orbitals on Co(tet) in the minority spin state. In Table 5,
the total and partial DOS at the Fermi level (inverse type B) are reported for GGA and QSGW.
The total DOS is similar to each other and the main contribution is from Co(tet). However, there
are some remarkable diﬀerences in its component. The components of Ni(oct), Co(oct), and O
decreases from GGA to QSGW, whileas the component of Co(tet) increases. These changes on
Table 2: Atomic and total magnetic moments (𝜇B ) in NiCo2 O4 (inverse type B) from GGA, oneshot GW, and QSGW. Note that the averaged value is presented at O 2p.
Atom
Co(tet) 3d
Ni(oct) 3d
Co(oct) 3d
O 2p
Sum in f.u.
Total in f.u.

GGA
2.34
−0.87
0.09
0.07
1.81
2.00

one-shot GW
2.77
−1.27
0.02
0.09
1.86
2.00

QSGW
2.77
−1.39
0.07
0.10
1.86
2.00

Table 3: Atomic and total magnetic moments (𝜇B ) in the normal and inverse A spinels of NiCo2 O4
from QSGW. Note that the averaged value is presented at O 2p.
Atom
Co(tet) 3d
Ni(tet) 3d
Ni(oct) 3d
Co(oct) 3d
O 2p
Sum in f.u.
Total in f.u.

Normal
1.56
0.00
0.09
1.92
2.00

Inverse A
2.76
1.56
−1.31
0.03
0.08
1.86
2.00

Table 4: The number of electrons in the projected orbitals in NiCo2 O4 (inverse type B) from GGA,
one-shot GW, and QSGW. u+d speciﬁes the sum of up and down electrons.
Atom
Co(tet) 3d
Ni 3d
Co 3d
O 2p

up
4.56
3.37
3.46
1.74

GGA
down
2.22
4.25
3.38
1.67

u+d
6.77
7.62
6.84
3.40

one-shot GW
up down u+d4.67 1.89 6.56
3.13 4.40 7.52
3.36 3.34 6.70
1.73 1.64 3.36

up
4.66
3.08
3.38
1.74

QSGW
down
1.89
4.46
3.31
1.64

u+d
6.55
7.54
6.69
3.37

Hasan Al Rasyid, Masao Obata, Indra Pardede, Marleni Wirmas, Takao Kotani, Tatsuki Oda

11

Table 5: Total and projected DOS 𝑁 ↓ at 𝐸 F (states/eV/cell) in inverse B from GGA and QSGW. At
the row of ”O 2p”, all O 2p contributions are included and at the row of ”Sum”, the summationsof atomic values are reported.
𝑁 ↓ (𝐸 F )Co(tet) 3d
Ni (oct) 3d
Co (oct) 3d
O 2p
Sum
Total

GGA
2.45
0.92
0.82
0.82
5.01
5.67

QSGW
3.50
0.16
0.42
0.53
4.61
6.15

Figure 7: Fermi surfaces in inverse type B spinel of NiCo2 O4 from QSGW.
the component imply a large change at the electronic state of the Fermi level.
From band dispersion curves in Figure 3, two bands cross the Fermi level at the minority
spin state. As a consequence, there are two Fermi surfaces, as depicted in Figure 7. As shown, theFermi surface is not so large that they extend to a large volume in the ﬁrst Brillouin zone (k-space).

3.5

Comparison with LDA+U

The LDA+U approach has been used in the calculation of electronic structures for a lot of oxidematerials. The necessary 𝑈eﬀ parameter is usually extracted from the empirical data. The recent
work by Shi et al., in which such parameters were determined in a ﬁrst-principles approach, is arare case. It can be stressed that QSGW is a non-empirical approach. Here, we point out several
diﬀerences with the LDA+U results for NiCo2 O4 . The LDA+U provides a set of merged bands[2, 5, 25], while the QSGW provides a set of separated bands, for example, at the unoccupied statein the majority spin state (see Figure 4) and at the occupied oxygen state. On Ni(oct), the energy
splitting between 𝑡2g and 𝑒 g seems to be diﬀerent between the majority and minority spin states.
As described, there are two energy gaps at the majority spin state around 𝐸 F which come fromthe 𝑡2g − 𝑒 g splittings on Ni(oct) and Co(oct). This consequence allows us to explain the twobandgaps extrapolated from the optical absorption spectra [3]. The edge-to-edge and peak-topeak energy diﬀerences are 1.26 eV and 2.11 eV for Co(oct) and 2.37 eV and 3.10 eV for Ni(oct).
These are comparable to the experimental gaps of 2.58 eV and 3.58 eV. Further correspondence
should be veriﬁed by advanced theoretical or experimental approaches.

12Electronic structure of spinel NiCo2 O4 : QSGW method

Despite small crystal ﬁeld splitting on the tetrahedral site (see Figures 4 and 6), the QSGW
result indicates two peaks in the Co(tet) 𝑒 g for both spin states. This new splitting implies somekind of eﬀective interactions between the cations at two neighboring tetrahedral sites (separated
by 3.5Å). As possible speculation, the QSGW may capture such intermediate-range interaction.

4

Summary

In summary, we have performed the ﬁrst-principles investigation on electronic structure in theferrimagnetic half-metallic inverse spinel nickel cobaltite NiCo2 O4 using a QSGW method. The
GWA of QSGW was found to conduct the narrower bandwidths in the band mainly consisting of
the cation 3d-orbitals, implying localized wave functions in the electronic structure. The energy
level splitting of 𝑡2g and 𝑒 g caused by its crystal ﬁeld from the PDOS results is more clearly observed than those in the GGA case. The QSGW method enables us to provide an alternative explanation for two kinds of band energy gaps from optical absorption spectra. This feature originatesfrom the QSGW to give a distinct crystal ﬁeld for diﬀerent cations, particularly the splitting of𝑡2g − 𝑒 g on Ni(oct) that is larger than that on Co(oct). These two gaps qualitatively make a contrast
with the single gap provided by GGA and LDA+U results. The main contribution to the states at
the Fermi level is found to be from Co(tet) 3d 𝑒 g orbitals. The possible states of Ni(oct) 𝑒 g , Co(oct)𝑒 g and others are largely suppressed. As in the previous calculations of GGA and LDA+U, theelectronic structure calculation from QSGW also conﬁrms that the material NiCo2 O4 is a promising candidate for spintronics application due to its half-metallicity. Unfortunately, the present
calculation fails to describe an insulating/semiconducting property inherent to the NiCo2 O4 at low
temperatures. This new problem should be solved with a more accurate treatment in an advanced
electronic structure calculation.

Acknowledgements
The computation in this work was partly performed using the facilities of the Supercomputer Center, Institute for Solid State Physics (ISSP), University of Tokyo, Japan. The authors (H.A.R., I.P.,
and W.R.) acknowledge the Japanese Government (MEXT) Scholarship in the Program for theDevelopment of Global Human Resources for Kanazawa University. The authors (T.O. and M.O.)
acknowledge the support from the Creation of new functional Devices and high-performance Materials to Support next-generation Industries (CDMSI), Japan.

References[1] B. Chi , H. Lin, J. Li, N. Wang, and J. Yang, Int. J. Hydrogen Energy 31, 1210 (2006).
[2] P. Li C. Xia, D. Zheng, P. Wang, C. Jin, and H. Bai, Phys. Stat. Sol. (RRL) 10, 190 (2016).
[3] Y. Bitla, Y. Chin, J. Lin, C. N. Van, R. Liu, Y. Zhu, H. Liu, Q. Zhan, H. Lin, C. Chen, Y. Chu, and Q. He, Sci.
Rep. 5, 15201 (2015).
[4] G. Blasse, Philips Res. Rept. 18, 383 (1963).
[5] K. Zhang, C. Zhen, W. Wei, W. Guo, G. Tang, L. Ma, D. Hou, and X. Wu, RSC Adv. 7, 36026 (2017).
[6] O. Knop, K. I. G. Reid, Sutarno, and Y. Nakagawa, Can. J. Chem. 46, 3463 (1968).
[7] P. Silwal, L. Miao, I. Stern, X. Zhou, J. Hu, and D. H. Kim., Appl. Phys. Lett. 100, 032102 (2012).
[8] K. Dileep, B. Loukya, P. Silwal, A. Gupta, and R. Datta, J. Phys. D: Appl.Phys. 47, 405001 (2014)[9] D. Santos-Carballal, A. Roldan, R. Grau-Crespo, and N. H. de Leeuw, Phys. Rev. B 91, 195106 (2015).

Hasan Al Rasyid, Masao Obata, Indra Pardede, Marleni Wirmas, Takao Kotani, Tatsuki Oda

13[10] S. V. Faleev, M. van Schilfgaarde, and T. Kotani, Phys. Rev. Lett. 93, 126406 (2004).
[11] M. van Schilfgaarde, T. Kotani, and S. V. Faleev, Phys. Rev. Lett. 96, 226402 (2006).
[12] T. Kotani, M. van Schilfgaarde, and S. V. Faleev, Phys. Rev. B 76, 165106 (2007).
[13] W. Kohn and L. J. Sham, Phys. Rev. 140, A1133 (1965).
[14] J. Otsuka, T. Kato, H. Sakakibara, and T. Kotani, Jpn. J. Appl. Phys. 56, 021201 (2017).
[15] A. Sawamura, J. Otsuka, T. Kato, and T. Kotani, J. Appl. Phys. 121, 235704 (2017).
[16] H. Sakakibara, T. Kotani, M. Obata, and T. Oda, Phys. Rev. B 101, 205120 (2020).
[17] The option of spin-orbit coupling was not introduced here: T. Oda and A. Hosokawa, Phys. Rev. B, 72, 224428(2005).
[18] M. G. Brik, A. Suchocki, and A. Kamińska, Inorg. Chem. 53, 5088 (2014).
[19] T. Kotani, J. Phys. Soc. Jpn. 83, 094711, (2014).
[20] T. Kotani, H. Kino, and H. Akai, J. Phys. Soc. Jpn. 84, 034702 (2015).
[21] O. Tange, GNU Parallel 2018, ISBN: 9781387509881 (2018).
[22] K. Momma and F. Izumi, J. Appl. Crystallogr. 44, 1272 (2011).
[23] A. Kokalj, J. Mol. Graph. Model. 17, 176 (1999).
[24] J. F. Marco, J. R. Gancedo, M. Gracia, J. L. Gautier, E. I. R 鱈 os, H. M. Palmer, C. Greavesc, and F. J. Berryd, J.
Mater. Chem. 11, 3087 (2001).
[25] X. Shi, S. L. Bernasek, and A. Selloni, J. Phys. Chem. C 120, 14892 (2016).