- Email: [email protected]
Electronic structure of CoO
ELSEVIER
Physica B 212 (1995) 125-138
Electronic structure of CoO Haoping
Zheng 1
Department of Physics and Astronomy, Louisiana State University. Baton Rouge, LA 70803, USA Received 28 July 1994; revised 24 October 1994
Abstract
The electronic structure of CoO is studied by a self-consistent cluster-embedding method. The localized and band properties are found in both ground state and excited states. An explanation for the insulating nature of CoO is proposed which is consistent with the one for NiO proposed previously. The simulations of both the spin-ordered states and the spin-disordered states (with local antiferromagnetic order) give a good description of the magnetic properties of CoO. It is found theoretically that CoO remains as an insulator when it goes from the paramagnetic phase to antiferromagnetic phase.
1. Introduction For more than five decades, the insulating nature of the transition-metal monoxides has been a continuing problem.. Due to historical reasons, NiO and CoO are considered as the representative transition-metal monoxides. In our previous paper [1], we reported the calculated electronic structure of NiO by using a self-consistent cluster-embedding (SCCE) method. The concept of double energy 9aps has been proposed to explain the insulating nature of NiO, which finds much experimental support. In this paper, the electronic structure calculated for CoO is presented. We show that the insulating nature of CoO can also be explained by the concept of double energy gaps. Experimental results show that NiO and CoO have similar properties. Both have the sodium chloride structure. Both show an antiferromagnetic transition at temperatures of 525 and 289 K, respectively. NiO and CoO are excellent insulators in
~Present address: Physics Department, Tongji University, Shanghai 200092, China.
both paramagnetic and antiferromagnetic phases. The optical-absorption coefficient of CoO is similar to that of NiO [2]. Similar photoemission spectra have been observed for NiO and CoO [3-5] However, compared with NiO, the situation in CoO is much less clear. Spin-polarized energy band calculations have obtained a small energy gap for NiO [6-9]. But CoO has an odd number of electrons per unit cell, which makes it more difficult for the band theory to reconcile its insulating nature. Even using the Slater approach to antiferromagnetism (each unit cell contains two CoO molecular units so that the number of electrons per cell is even), the results of spin-polarized energy band calculations still show a metallic nature for CoO in both paramagnetic and antiferromagnetic phases [6,9]. The self-consistent cluster embedding (SCCE) method was originally proposed by Ellis et al. [10] in 1977. Recently, we have proved that under certain conditions, the SCCE method separates the cluster-electrons from the background-electrons without changing the total energy of the system [ 1]. We call this electron ehar#e renormalization, which preserves the localized and band properties of the electrons in an embedded cluster. In addition, we
0921-4526/95/$09.50 (~i 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 1 100-1
H. Zheng / Physica B 212 (1995) 125-138
126
have developed a procedure to simulate the magnetic structure by using real atoms and ions, which makes this method useful in studying the magnetic properties of both spin-ordered and spin-disordered states [11]. As in the case of NiO, the calculation on CoO shows success: the properties of CoO can be interpreted naturally. The organization for the rest of this paper is as follows. Section 2 describes our theoretical model. Section 3 explains our computational procedures. The electronic structure of CoO is described in Section 4. Section 5 summarizes the results.
In a crystal, p2(r) can be obtained by using the crystal periodicity. Here we have introduced the orthogonality constraint, Vor, which is applied to the cluster-electrons in the surrounding core areas: M2
For :
Zi 2~j=l I r - e j l
if r is in the core of surrounding atoms,
0
otherwise.
(5)
During the calculation, Vor keeps the Pl out of the core area of the surrounding atoms. So within certain precision, we have
Pl (r) Vor(r) = 0
(6)
2. Theoretical model
A brief description of our theoretical model of the SCCE method is as follows. For a cluster of MI atoms (charge density Pl) embedded into an atomic array of M2 atoms (charge density P2), the total ground-state energy of the system (MI + M2 = M atoms) can be written [,12, 13] as
E~[p] = T[p] + EIc[P] + jfj f p(r)p(r') ~Z~7~ drdr' M
- 2 ~
~
p(r)Zi •
. ~
,--ur-e
~
i=1 J Ir - Ri[
ij
ZiZj
I~-RjI'
(1)
id:j
where p = p~ + P2. T[,p] is the kinetic energy of the noninteracting electrons, and Exc I-p] is the exchange-correlation energy. Ri and Zi are the position and charge of i'th nucleus, respectively. Atomic units are used here (e 2 = 2, h = 1, 2me = 1). Suppose we already know P2, by using the forms Pl (r) = p~P(r) + p~"(r) = ~
~
[~bf (r)l 2,
(2)
o occupied I
T[,pl] -- ~
~,
fq~'(r)(-V2)c~(r)dr,
(3)
occupied I
we derive the basic formula of the SCCE method [1]: { _ VZ + 2 fpl(r')_+ p z ( r ' ) d r ' - 2
J
Ir-r'l
~ Z, i=1 I r - e i l
+ V ~c(r) + Vo~ ~P7(r) = 27 ~b7(r). J
(4)
everywhere. This means that, although Eq. (4) contains an artificial potential Vor, if the condition (6) is satisfied with high precision, begin from Eq. (4), we can still derive the Kohn-Sham ground-state energy functional (1) which contains no Vor. Thus the separation of the cluster-electrons from the background-electrons does not change the total energy of the system. According to extended Hohenberg-Kohn theorem [14-17], unchanged total ground-state energy may mean unchanged total electron charge density. So the electron charge density calculated from Eq. (4) should be close to the real one and the embedded cluster can represent a portion of a solid. We call this electron charge
renormalization. In a solid, a band electron (here we mean it is shared by all atoms in the crystal) is usually described by Bloch plane waves. It is difficult to describe such a band electron directly by a finite cluster. What we can do is to try make an electron charge renormalization, which restricts the electrons around the cluster under the condition that the total ground-state energy and the total electron charge density are unchanged. Then it is reasonable to expect that both localized and band properties of the electrons are preserved in an embedded cluster, which means a band electron is now shared only by the atoms in the cluster, and a localized electron is still localized to the same atom. The total ground-state energy Ec is proportional to the number of cells. We have introduced a formula to represent the energy of an embedded
O1T !CO2
H. Zheng / Physica B 212 (1995) 125-138
cluster [1]:
E1 [N1, M,] = T[p,] + Ex¢, [p,,p2] +
i ~Pl(r)pl(r') a a ' f fP' (r)p21rt) j -~-_-r7] urur + j j ir--r-;I drdr' M
I
127
[ Pl (r)Zi d r , 2/='~ 1~ J [r -- Ril
COIL.) (7)
where EI~I [Pl, P2] = ~ P~ ex~(pup, pdn)dr. Fixed nuclei were assumed and the Coulomb interaction energy between the nuclei was removed. Definition (7) makes this method useful in the study of magnetic structure.
3. Computational procedure A general program package for an embedded cluster system using Gaussian orbitals has been developed by the author. Eq. (4) is solved self-consistently. The computational procedure, geometrical arrangement and embedding procedure used here are the same as that of the embedded Ni202 cluster [1]. Only a brief description is given here. Fig. 1 shows the four-atom Co202 cluster with the lattice constant 4.2667 ,A, the value of the lattice constant in crystalline CoO [18]. The number of surrounding atoms is 124. The number of surrounding point charges is 5380, with + 21el charge in Co sites and -21el charge in O sites. On the outmost boundary, fractional (½, ¼, and ~) charges are put on the faces, edges, and corners. This makes the Madelung constant at the positions of cluster atoms equal to 1.747565, which is the same as the exact Madelung constant of an ionic crystal with the sodium chloride structure [19]. The appropriateness of this embedding arrangement has been tested in the calculation of NiO. We use the von Barth and Hedin [16] form of the exchange-correlation potential, as parameterized by Rajagopal and co-workers [20]. A total of 240096 points is used in the Vxc calculation. In order to solve Eq. (4), the single-particle wave functions are expanded into a set of Gaussian orbitals. Table 1 gives basis set for CoO which was originally provided by Wachters [21] and van Duijneveldt [22]. A d polarization function is in-
02
Fig. 1. Four-atom Co202 cluster. cluded in the basis of oxygen [23]. Compared with the original bases, eight exponents have been changed, four exponents have been inserted and 19 diffuse exponents have been added. The ratios between changed and added adjoining exponents are about 2. A test of the basis set in two-atom-cluster calculation showed satisfactory convergence. The orthogonality constraint Vor of Eq. (5) is evaluated numerically in each of the surrounding atoms and point charges. The atomic core cutoff radius ro is adjusted according to two simple rules: (1) avoid numerical instability; (2) keep the total energy minimum. The results are 0.387 a.u. for cobalt atom and 1.132 a.u. for oxygen atom. In the calculation of Vor, 1376 points and 2494 points are used for the cobalt and oxygen cores, respectively. A two-step method is used because of the lack of disk-storage space. In the first step, a two atom Co-O embedded cluster is calculated with the selfconsistent determination of the surrounding charge density. In the second step, a four-atom Co202 embedded cluster is calculated with fixed surrounding charge density P2. Initially, Pe is built from Pl of the first step. Then t92 is built from p~ of the new results of four-atom Co202 cluster again and again. When the differences of all eigenvalues are less than 0.0002 Ry, and the difference of the total energy is less than 0.0007 Ry, the iterations were stopped. All background charge densities, P2, are built from the same Pl and are fixed during the calculation. The SCF iteration is only with respect to p~.
4. Results for an embedded Co202 cluster 4.1. Cluster ground state We build P2 from Pl according to the antiferromagnetic structure [24]. After 60 SCF iterations, the
H. Zheng / Physica B 212 (1995) 125-138
128 Table 1 Original
Basis set of cobalt atom s(11) Same Same Same Same Same Same Same Same Same Same Same Same 0.122266 0.044172
d(12) Same Same Same 1.4433 0.449965
Basis set of oxygen atom S(II) Same Same Same Same Same Same Same Same Same Same Same Same Same
d(1) Same
O u r basis
Original
Exponent
Coefficient
270991.0 39734.8 9057.46 2598.21 868.200 323.431
3.100D - 04 2.420D - 03 1.238D-02 4.849D-02 1.467D-01 3 . 2 1 3 D - 01
Our basis Exponent
Coefficient
1636.21 390.903 127.884 49.2413 20.7512
2.9600D 2.3360D 1.0343D 2.7954D 4.3268D
p(lO)
130.860 56.1219
4.0497D - 01 1.9961D - 01
18.9219 7.95238 2.19754 0.846713 0.205000 0.129199 0.060093 0.027950 0.013000
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
44.9774 12.5690 4.24422 2.20180 1.142242 0.592568
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
105374.95 15679.240 3534.5447 987.36516 315.97875 111.65428 42.699451
1 . 4 3 0 0 D - 04 1.1230D - 03 5 . 9 8 0 0 D - 03 2 . 5 5 6 2 D - 02 9 . 2 5 9 0 D - 02 2.81749D-01 6.77164D-01
17.395596 7.4383090 3.2228620 1.2538770 0.49515500 0.19166500 0.09043500 0.04267100 0.02013400 0.00950000
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1.154000
1.0000
Same Same Same Same Same Same Same Same Same
0.1219
p(13) Same Same Same Same 2.102525 0.850223 Same 0.128892
9.20368 3.81779 1.58762 0.624660 0.170440 0.083140 0.040550 0.019780 0.009650
1.0000 1.13000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
0.307410 0.159477 0.082733 0.042920 0.022266 0.011551
1.0000 1.0000 1.0000 1,00130 1.0000 1.0000
200.00000 46.533367 14.621809 5.3130640 2.6675250 1.3392810 0.6724120 0.3375970 0.1713690 0.0869890 0.0441570 0.0224150 0.0113780
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
-
03 02 01 01 01
H. Zheng / Physica B 212 (1995) 125 138
total energy has converged within 10-7 Ry. There are 0.00095 cluster-electrons remaining in the surrounding core regions, which means the condition (6) is well satisfied. Table 2 gives the eigenvalues and the Mulliken populations of spin-up orbitals. The spin-down orbitals are the same as the spin-up orbitals, except for the exchange of atoms 1 and 2. The Mulliken population analysis shows that each cobalt 3d orbital is mostly attached to a particular cobalt ion, while each oxygen 2p orbital is shared by two oxygen ions. Here we introduce the concept of charge overlap between two orbitals i and j: (Charge overlap)/j = f Minimum (pi(r), pj(r)) dr. (8) Unless otherwise specified, the "overlap" in this paper means charge overlap. The overlap between the two orbitals # 35 (spin-up and spin-down, both are cobalt 3d electrons but attached to different cobalt atoms) is 3.16%. In contrast, the overlap between the two orbitals # 28 (oxygen 2p electrons) is 84.97%. The overlap between the two orbitals # 39 (oxygen 3s electrons) is 92.52%. The picture of the ground state is the coexistence of localized and band properties: a 0.13 eV energy gap separates the unoccupied and occupied cobalt 3d orbitals which are well localized (this energy gap is too small to make CoO an insulator and will be discussed in Section 4.4). Each 3d orbital is mostly attached to a particular cobalt ion. Below the 3d levels are two diffuse oxygen 2p bands; above the 3d levels are empty oxygen 3s, cobalt 4s and oxygen 3p bands. The Mulliken population analysis show that the spin magnetic moment of cobalt ion is 2.94~B. The experimental data for the total magnetic moment of cobalt ion in CoO (including orbital contribution) is 3.35/~B [25], 3.8/~B. [26] and 3.8/~B [27]. The X-ray absorption spectroscopy shows that for Co 2 + ion the contribution of orbital-moment and spin are about 0.5-0.8/~ and 2.8/~B, respectively [28]. The Co202 clusters is in an antiferromagnetic state. Two cobalt ions have opposite spin magnetic moments, and two oxygen ions have no spin magnetic moment, so the total magnetic moment of the cluster is zero. This is the result of 90 ° super ex-
change effect.
129
4.2. Optical properties Fig. 2(a) shows the normal-emission angle-resolved photoemission experimental data and the results of energy-band calculations of CoO reported by Shen et al. [4]. Fig. 2(b) shows our calculated eigenvalues. For the reasons discussed in the previous paper [1], we simply use the eigenvalues in Table 2 as the ionization potential. Unfortunately, no off-normal-emission experimental data are available. For NiO, the off-normal-emission experimental data show more features than the normal-emission data [5]. Some features may be missing in Fig. 2(a) because of the rigorous selection rules in a normal-emission experiment. In general, our results are similar to the energy-band calculation. Both show that cobalt 3d bands are above the oxygen 2p bands, which is in agreement with the experimental data. The angle-integrated photoemission data of CoO show a satellite located about 8.5 eV lower than the first peak [4]. Shen et al. assigned it as a d 6 satellite and use it to derive a 9-11 eV Ud value. In our calculation, no such satellite appears. As Co 3d electrons are well localized and strongly correlated, we think the satellite may come from the ionization of Co 3+ ion following the ionization of Co 2 + ion (double holes). No confirmation has been made since we are unable to calculate a full relaxed ionization state. Our high-quality basis set and relatively accurate embedding procedure make the lowest several unoccupied orbitals acceptable, which enables us to interpret the experimental optical-absorption data. Fig. 3(a) shows the experimental result for the optical-absorption coefficient of CoO obtained by Powell and Spicer [-2]. Fig. 3(b) shows the optically allowed transition energies obtained from the differences of the calculated eigenvalues. Following the discussion of Ref. [1], the real value will be a little larger than that in Fig. 3(b) if the transition involves a large electron rearrangement. Included are all optically allowed transitions, from cobalt 3d and oxygen 2p valence orbitals to the unoccupied cobalt 3d, oxygen 3s, cobalt 4s and oxygen 3p orbitals. The highest unoccupied orbital counted is an oxygen 3p orbital with eigenvalue of 0.5598 Ry (7.6 eV). Part A of Fig. 3(b) contains the energies of the charge transfer transitions from oxygen 2p orbitals to the unoccupied cobalt 3d orbitals. Because the
H. Zheng / Physica B 212 (1995) 125 138
130
Table 2 E i g e n v a l u e s a n d Mulliken p o p u l a t i o n s of C o 2 0 2 cluster in g r o u n d state (spin-up orbitals) Orbital
No.
O 3p
44
0.5598
0.01
- 0.01
Co4p
43
0.5175
0.50
- 0.04
0.01 - 0.04
Co4s
42
0.4161
0.78
0.03
- 0.03
-
0.3852 0.2290
1.01
- 0.25 - 0.29
- 0.04 - 0.06
- 0.04
O 3s
41 40
- 0.06
1 . 2 5
39
0.1256
0.05
- 0.15
- 0.02
- 0.02
1.13
38 37
- 0.2014 - 0.2543
0.02
36
- 0.2830
C o 3d
E n e r g y (Ry)
Co s
Co p
Co 1 d
Co 2 d
O 1p
O 2p
0.50
0.50
0.29
0.29
0.23 0.16
0.23 0.16
0.08
0.08
0.01
0.01
0.98
0.01
0.01
0.97 0.02
0.01 0.01 0.04
0.01 0.01 0.04
0.01
0.01
0.01 0.03
0.01
O s
- 0.23 -
0.95 0.98
-
0.01
0.02 0.01
Note: A b o v e are the u n o c c u p i e d orbitals C o 3d
O2p
O 2s C o 3p
C o 3s
35(Ef) 34
- 0.2926 - 0.3277
33
- 0.3959
32 31
- 0.4525 - 0.4766
0.01
30 29
- 0.4873 - 0.5111
0.02
28 27 26
Co2p
C o 2s C o ls
0.01
0.97 0.01
0.89 0.97 0.94
0.01
0.94 0.84
0.01 0.01
- 0.6190
0.16
- 0.6350 - 0.6906
0.06
0.02 0.01
0.05
25
- 0.7069
0.01
0.08
24
- 0.7103
0.17
- 0.07
0.04
23
- 0.7612
0.14
0.05
0.01
22 21 20 19
-
1.6875 1.6952 4.3645 4.4139
- 0.05
- 0.03 - 0.03 1.00 1.00
18
- 4.4364
1.00
17
- 4.5901
1.00
16
- 4.6347
1.00
15 14
- 4.6570 - 6.9198
1.00
- 7.1479
1.00
13 O ls
0.01
12 11 10 9 8 7
-
6 5 4 3 2 1
-
0.03 0.02 0.08 0.47
0.47
0.45
0.45
0.06
0.43 0.43 0.44
0.43 0.43 0.44
0.01 0.01
0.01 0.01
- 0.08 1 . 0 3
1.07
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
0.41
0.02 0.08 0.41
0.03
1.00
37.2358 37.2358 55.9840 55.9892 55.9913 56.0769
56.0816 56.0837 64.3784 64.5002 - 551.0222 - 551.0232
0.02
-
H. Zheng / Physica B 212 (1995) 125-138
Co 0 3d 2p
'
i
'
I
'
I
'
C o O (001)
~
I
~
vv
t6
'
CoO ABSORPTION
.~ 14
-
131
-
COEFFICIENT
E
0 ~J
12
I0 PI36
>•
~08
-4
g •
~5 E ~
PRO
&5
0
~o4 -6 O~
o
p 2
i 4
6
(a) o
o
~
c
v
A
I 0.2
,
(b)
1
[ 0.4
,
I 0.6
C~slal Momentum
(b)
, I0 12 14 16 IR PHOTON ENERGY {eV}
~ ~
0 ,
8
,
I
,
0.8
B
X
cobalt 3d orbitals are well localized and oxygen 2p orbitals are diffuse, transfering one oxygen 2p electron to an unoccupied cobalt 3d orbital will cause large electron rearrangement. The real transition energies should be a little larger than that in part A. Considering this factor, part A is reasonably in agreement with the experimental data. The smallest transition energy in part A is 4.75 eV. We estimate that the main optical absorption edge is about 5 eV. Parts B, C, and D in Fig. 3(b) contain the transition energies of O 2p ~ O 3s, O 2p ~ Co 4s, and Co 3d ~ O 3p (Co 4p), respectively. In general, they are lower by about 2 eV than the experimental data. We think the error comes mainly from the inaccuracy of the unoccupied O 3s, Co 4s and O 3p (Co 4p) orbitals. In cluster calculations, usually, the higher the unoccupied orbital is, the less accurate is its energy. Although they have the correct order (by which we mean they are 'acceptable'), the orbitals of O 3s, Co 4s and O 3p (Co 4p) are certainly less accurate than the unoccupied Co 3d orbitals. Considering this factor, we interpret that the peak
22
24
26
c
III I1 Illl [nml D
(a)
Fig. 2. (a) The photoemission experimental data and the results of energy-band calculations of CoO [4]. (b) The eignevalues of embedded Co~O2 cluster. The position of the first Co 3d orbital in (b) is aligned to the first experimental data in (a) (represented by V).
, 20
Fig. 3. (a) The experimental results of optical-absorption coefficient of CoO [2]. (b) The optically allowed transition energies obtained by using the eigenvalues.
around 13.6 eV comes from part B; the peak at 18.4 eV comes from part C and part D (part D involves a large electron rearrangement so the real energies should be even larger). The weak optical-absorption peaks below the threshold are the 3d-3d dipole-forbidden transitions of the same cobalt ion. Johansen and Wahlgren [29] have calculated these d-d excitation spectra by using SCF/CI techniques on a CoO6 10 cluster. In our calculation, we can get twelve 3d 3d transition groups by using the eigenvalues in Table 2. Their average values are (in eV) 0.13, 0.57, 1.00, 1.24, 1.63, 1.93, 2.31, 2.69, 3.10, 3.46, 3.82 and 4.21. The experimental data between 0.7 and 2.9 eV are about 0.83, 0.90, 1.03, 1.24, 1.63, 2.03, 2.30, 2.49 and 2.61 eV [30]. Finally, by using the eigenvalues, the cobalt 3p to 3d threshold is 56eV, close to the experimental result of about 60 eV.
4.3. Magnetic properties This subsection describes the simulation of spin disordered states with local antiferromagnetic (AF) order of CoO.
H. Zheng / Physica B 212 (1995) 125-138
132
First, we estimate the stability of the local antiferromagnetic (AF) order. We calculated the total energy of the ferromagnetic state. The result is 3.51 eV (41000 K) above the energy of the AF state. This indicates very stable local AF order. So it is reasonable to assume that in the paramagnetic state, local AF order still remains and the local moments of cobalt ions persist essentially unchanged. In order to simulate the spin disordered states with local AF order, two geometrical restrictions have been applied: (1) the total number of spin-up cobalt ions should equal the total number of spin-down ions; (2) in any cube (containing four cobalt ions and four oxygens ions), the four cobalt ions should not have the same spin direction. A random-number-generating subroutine is used to choose the spin-up and spin-down atoms, then the two geometrical restrictions mentioned above are applied to the 18 nearest Co ions (the change of spin direction of the other Co ions will not change the result).
Table 3 gives our results. The last two rows show the special cases which violate our restrictions. One row has all spins up, the other has each clustercobalt ion surrounded by spin-parallel cobalt ions. Both cases cannot happen in practice because of the local AF order. Apparently, the AF structure has the lowest energy among the disordered structures. Because of the local AF order, the disordered states are actually not completely disordered, so the change of the system entropy may be small. If we neglect the change of system entropy, by averaging the eight disordered results in Table 3, we get the N6el temperature of 1162 K; the experimental result is 289 K [31]. We think the discrepancy comes mainly from the imperfection of the exchange-correlation potential we used. Table 3 also shows the energy gaps, the optical absorption edges and the spin magnetic moment of each cobalt ion. It is clear that they are independent of the long-range spin order. The calculated spin
Table 3 Results of a four-atom CO202 cluster with different atomic spin direction arrangement of surrounding atoms (the Co202 cluster is in antiferromagnetic state) No.
Atomic spin direction arrangement
18 nearest Co atoms up
1 2 3 4 5 6 7 8 9 10
Antiferromagnetic (31 up, 31 down) Disordered 1 (31 up, 31 down) Disordered 2 (31 up, 31 down) Disordered 3 (31 up, 31 down) Disordered 4 (30 up, 32 down) Disordered 5 (31 up, 31 down) Disordered 6 (31 up, 31 down) Disordered 7 (31 up, 31 down) Disordered 8 (31 up, 31 down)
All spins up
Total energy
Energy gap
Absor, edge
Spin magnetic moment (/~B)
(K °)
(eV)
(eV)
Co 1-up
Co 2-down
down
9
9
0
0.130
4.572
2.9397
2.9397
9
9
+ 1202
0.102
4.545
2.9355
2.9379
9
9
+ 1587
0.100
4.539
2.9370
2.9375
9
9
+ 704
0.128
4.570
2.9393
2.9380
9
9
+ 711
0.119
4.559
2.9399
2.9377
9
9
+ 1444
0.106
4.562
2.9371
2.9393
9
9
+ 1802
0.100
4.547
2.9373
2.9371
9
9
+ 399
0.120
4.560
2.9402
2.939
9
9
+ 1446
0.103
4.546
2.9364
2.9379
18
0
+ 1383
0.149
4.599
2.9366
2.9408
9
9
- 1173
0.152
4.590
2.9412
2.9412
(62 up, 0 down) 11
Special case (31 up, 31 down)
H. Zheng/Physica B 212 (1995) 125 138
magnetic moments of cobalt ions in spin disordered states are almost the same as that in the AF state, which shows the consistency of our assumption. Table 4 shows the eigenvalues and the Mulliken populations of one of the spin-disordered states (disordered 1 in Table 3). Tables 4 and 2 are almost identical, which means they have similar electronic structure. Actually, we find that the electronic structure of the cluster remains almost the same in all spin-disordered states as in the AF state. This means that in a local AF pair, the orbitals are nearly the same for both paramagnetic and AF phases. The photoemission experimental data support our results: the photoemission spectra taken in the two phases of CoO are identical [4]. From the discussion of the next section, we can say that CoO remains as an insulator when it transforms from the AF to paramagnetic phase. This result is in agreement with the observed properties of solid CoO.
4.4. Energy gap In this section, the same idea and method as in the case of NiO [1] are used to give an explanation of the insulating nature of CoO. It is clear that our energy gap of 0.13 eV cannot make CoO an insulator. From Table 5(a), we know the overlap of the highest occupied 3d orbital (orbital 35) is 3.16%, the overlap of the first unoccupied 3d orbital (orbital 36) is 1.97%. Using the idea and explanation of our previous paper [1], it is clear that these two orbitals cannot form metallic bands, so the ground state of cluster Co202 is an insulating state. Fig. 4(a) shows the charge distribution of the ground state along the line connecting Co 1 and O 1 ion. The charge distribution of the ground state along the line connecting two cobalt ions is shown in Fig. 4(b). It is clear that the electrons of cobalt ions are well localized and there is almost no overlap between the two cobalt ions. In order to discuss the insulating gap, the Hubbard U parameter for cobalt 3d electron is estimated first. For the embedded Co202 cluster with fixed P2, the ionization potential of the orbital 35 and the electron affinity of the orbital 36 (see Table 2) were calculated by using the transitionstate method. We take the point of view that
133
U should be the difference of ionization potential and the electron affinity. The results is Udd ~ l l.0eV. Because of limited electron rearrangement during the transition-state calculation (p2 fixed), the real Udd value should be smaller than that one. Janssen and Nieuwpoort [-32] and Martin [33] have reported the long-range polarization contribution which reduces the calculated Udd of a cluster dramatically. Unfortunately, we are unable to calculate the Udd with full relaxation. The best estimate we can make is that the Udd is probably in the range of 7-9 eV. The electronic structures of 13 excited states have been calculated. The results of all six 3d-3d transitions involving the same cobalt ion and the same spin direction (spin-preserving transitions) are presented in Table 5(a). These results are reliable because the electron rearrangement is very small during the transitions (compared with the ground state in Table 5(a), the changes of the eigenvalues are less than 0.015 Ry). In Table 5(a), the maximum overlap of an excited 3d electron is 2.750, the maximum overlap ofa 3d hole is 3.25%. Fig. 5 shows the charge distribution of state 2 along the line connecting two cobalt ions. Table 5(b) gives the results of six 3d 3d spin-flipped transitions between the orbitals of the same cobalt ion. The probabilities of these transitions are much lower than that in Table 5(a) because they are dipoleforbidden and they reduce the spin magnetic moment of cobalt ion to about 1/~B. The electron rearrangement in these transitions are larger than that in Table 5(a) due to the change of spin magnetic moment and the change of exchange-correlation energy. Nevertheless, we think it is acceptable to get the localized and band information from these results. Our experience shows that the calculations of spin-flipped transitions are convergent except the one from orbital 32 to 38. The transitions from the lowest occupied 3d orbital (orbital 29) to the three unoccupied 3d orbitals are calculated. Four of the five transitions from the occupied 3d orbitals to the highest unoccupied 3d orbital (orbital 38) are also calculated. We think it is sufficient for our argument, because the excited electron in orbital 38 and the hole in orbital 29 or 30 usually have the maximum overlap. In Table 5(b), the largest overlap of an excited 3d
H. Zheng / Physica B 212 (1995) 125-138
134
Table 4 E i g e n v a l u e s a n d M u l l i k e n p o p u l a t i o n s of C o 2 0 2 cluster in s p i n - d i s o r d e r e d 1 state (spin-up orbitals) Orbital
No.
O3p Co4p
44 43
0.5602 0.5170
Co4s
42
0.4169
0.80
O3s
41 40 39
0.3840 0.2291 0.1251
C o 3d
E n e r g y (Ry)
38
- 0.2030
37
-0.2559
36
- 0.2846
Co s
Co p
Co 1 d
Co 2 d
O s
O 1p
O 2 p
0.48 0.30
0.53 0.29
0.01
- 0.01
0.50
- 0.04
- 0.01 - 0.04
1.00
0.02 - 0.24
- 0.03 - 0.04
- 0.02 - 0.04
- 0.23 - 0.01
0.23 0.15
0.18
- 0.06 - 0.02
- 0.06 - 0.02
1.25 1.14
0.08
0.08
0.04
- 0.29 - 0.15
0.95
0.02
0.01
0.01
0.98
- 0.01
0.98
0.01
0.01
0.97
0.01 0.02
0.01 0.01
0.02
0.04
0.04
- 0.01
0.01 0.03 0.02
0.01 0.03 0.02
0.08 0.47
0.07 0.35
0.40
0.53
0.49 0.54
0.42 0.32
0.02
Note: A b o v e are the u n o c c u p i e d orbitals Co3d
O2p
O2s Co3p
Co3s O ls Co2p
Co2s C o ls
35(El)
- 0.2921
34 33
- 0.3271 - 0.3953
32
- 0.4516
31
- 0.4756
30
- 0.4873
29 28 27 26 25 24
-
0.5109 0.6186 0.6346 0.6896 0.7079 0.7104
23
- 0.7617
0.01
0.01
0.97 0.01 0.01 0.02
0.05
0.01
20 19 18
- 4.3671 - 4.4140 - 4.4363
1.00
17
- 4.5889
1.00
16 15 14
- 4.6346 - 4.6568 - 6.9206
1.00
- 56.0838 - 64.3788 - 64.5002 - 551.0226 -551.0233
0.01 0.02 0.01 0.02
0.14
- 1.6872
5 4 3 2 1
0.84 0.16 0.06 0.07 0.04
- 1.6949
-7.1474 37.2354 37.2364 55.9846 55.9895 55.9916 56.0768 56.0817
0.01
0.02 0.16
21
-
0.97 0.94 0.94
0.05 0.01 - 0.06
22
13 12 11 10 9 8 7 6
0.89
- 0.03 -0.05
- 0.03
0.01
0.03 0.06
0.27
0.58
- 0.08
0.46
0.43
1.03 1.07
0.01 0.01
0.01
1.00 1.00
1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00
H. Zheng/Physica B212 (199~ 125-138 ii-
10-
(a)
I0,.....,
9-
m t~ ¢xi o o o c~
8-
c5
5~
76-
I
s ......
[
9-
Co Co
p . . . .
l
8-
0
s
0
p
I
Co s . . . . . .
I
'q
7.
0
6.
o o
5-
to N
C:I
I
I
Co p - Co
d
li i
,:5 4-
4-
Z
135
E,-,
3.
z;
2-
¢:::1
1,
3
'\
/
Z 1
0
0 ~
I
0
'
I
1
Col
'
I
2
'
I
'
4
3
DISTANCE(a.u.)
01
I 1
Col
'
I
I
2
3
'
L 4
DISTANCE(a.u.)
'
I
'
5
Co2
Fig. 5. The charge distribution of excited state 2 along the line connecting two cobalt ions. 10
I
(b) s
Co
I
I
.......
8 to ¢kl O
I:
7 6-
I'
5-
I
o o
Z 1O. t 0
Col
1
2
3
4
DISTANCE(a.u.)
'
1
5 Co2
Fig. 4. The charge distribution of ground state 0fC0202 cluster: (a) along the line connecting Co 1 and O 1 ions; (b) along the line connecting two cobalt ions.
electron is 4.63%, the largest overlap of a 3d hole is 10.73%. The result of state (8) in Table 5(a) is not reliable because its large electron rearrangement is partially restricted by the fixed P2. The only certain conclusion we can draw from it is that each 2p hole
is shared by two oxygen ions. It is interesting to find that orbital 36 is still well localized (which means Co ÷ is localized), although we cannot confirm it absolutely. In conclusion, similar to the ground state, the excited electrons and holes have also both localized and band properties. For all spin-preserving 3d-3d transitions, the overlaps of excited orbitals are less than 2.8%, and the overlaps of holes are less than 3.3%. For spin-flipped 3d-3d transitions, the overlaps of excited orbitals may be less than 4.7%; the overlaps of holes may be about 11%. The holes in oxygen 2p orbitals are diffuse, which means the holes can move freely in solid. CoO has the following four properties (see Fig. 6): (1) the minimum energy required to transfer an electron from oxygen 2p bands to a cobalt site is about 5 eV (the main optical absorption edge); (2) the minimum energy required to transfer a cobalt 3d electron to the oxygen 3s conduction band is about 6 eV; (3) the minimum energy required to transfer an electron from one cobalt ion to another cobalt ion far away is about 8 eV (the Hubbard U parameter for cobalt 3d electron); (4) the 0.13 eV energy gap is between two localized cobalt 3d orbitals. The 3d excited orbitals and holes of cobalt ion are well localized.
H. Zheng / Physica B 212 (1995) 125 138
136
Table 5(a) Eigenvalues and Mulliken populations of C o 2 0 2 cluster in the g r o u n d state and seven spin-preserving excited states (spin-up orbitals) State
Orbital No.
Occupation number
Eigenvalue (Ry)
Co 1 (%)
Co 2 (%)
O 1 (%)
O 2 (%)
Overlap (%)
38 37 36 35 (Ef) 34 33 32 31 30 29 28
0 0 0 1 1 1 1 1 1 1 1
- 0.2014 - 0.2543 - 0.2830 - 0.2926 - 0.3277 - 0.3959 - 0.4525 - 0.4766 - 0.4873 -0.5111 - 0.6190
0.4 0.7 0.0 0.5 0.1 89.6 97.8 93.5 95.0 84.1 15.6
94.9 99.4 98.2 97.2 97.0 0.5 1.0 0.1 0.6 0.9 1.8
2.3 0.0 0.9 1.1 1.4 4.9 0.5 3.2 2.2 7.5 41.3
2.3 0.0 0.9 1.1 1.4 4.9 0.5 3.2 2.2 7.5 41.3
2.65 1.12 1.97 3.16 3.17 5.20 2.37 6.76 5.77 16.36 84.97
(2) 35 to 36
36(exc.) 35(hole)
1 0
- 0.2860 - 0.2944
0.0 0.5
98.2 97.3
0.9 1.0
0.9 1.0
1.95 3.05
(3) 35 to 37
37(exc.) 35(hole)
1 0
- 0.2516 -0.2893
0.7 0.5
99.6 97.1
- 0.1 1.2
- 0.1 1.2
1.05 3.16
(4) 35 to 38
38(exc.) 35(hole)
1 0
- 0.2031 - 0.2931
0.4 0.5
94.7 97.2
2.4 1.1
2.4 1.1
2.75 3.25
(5) 34 to 36
36(exc.) 34(hole)
1 0
- 0.2734 - 0.3168
0.0 0.1
98.3 97.2
0.9 1.3
0.9 1.3
1.88 2.93
(6) 34 to 37
37(exc.) 34(hole)
1 0
- 0.2399 - 0.3130
0.7 0.1
99.6 97.3
- 0.1 1.3
- 0.1 1.3
1.04 2.88
(7) 34 to 38
38(exc.) 34(hole)
1 0
- 0.1894 - 0.3146
0.4 0.1
94.9 96.9
2.3 1.5
2.3 1.5
2.68 3.23
(8)
36(exc.) 35 34 28(hole)
1 1 1 0
0.0160 - 0.0076 - 0.0327 - 0.8205
0.2 - 6.9 0.0 0.3
97.7 69.9 98.6 0.2
1.0 18.5 0.7 49.7
1.0 18.5 0.7 49.7
2.02
(1) ground state
28 to 36
Table 5(b) Eigenvalues and Mulliken populations of
Co202 cluster
State
Orbital No.
Occupation number
(9) 29 to 36
36(exc.) 29(hole)
(10) 29 to 37
99.86
in six spin-flipped excited states (spin-up orbitals)
Eigenvalue (Ry)
Co 1 (%)
Co 2 (%)
O 1 (%)
O 2 (%)
Overlap (%)
1 0
- 0.3343 - 0.4410
0.1 93.8
97.8 2.2
1.1 2.0
1.1 2.0
2.50 8.41
37(exc.) 29(hole)
1 0
- 0.3002 - 0.4362
1.3 94.0
98.5 2.1
0.0 1.9
0.0 1.9
2.57 8.11
(11) 29 to 38
38(exc) 29(hole)
1 0
- 0.2506 - 0.4389
0.8 92.8
93.4 2.6
2.8 2.3
2.8 2.3
4.31 9.73
(12) 30 to 38
38(exc.) 30(1101e)
1 0
- 0.2695 - 0.4292
0.9 93.6
92.9 4.2
3.0 1.1
3.0 1.1
4.63 10.73
(13) 31 to 38
38(exc.) 31(hole)
1 0
- 0.2629 -0.4089
0.9 96.2
93.0 0.5
2.9 1.6
2.9 1.6
4.58 4.52
(14) 33 to 38
38(exc.) 33(hole)
1 0
- 0.2606 -0.3266
0.9 92.3
93.5 1.3
2.8 3.2
2.8 3.2
4.30 5.04
H. Zheng/Physica B 212 (1995) 125 138
5. Summary
I O 3s conduction band
2 (6ev) 3 (8ev)
uC°~l~!dpied ( f ; ! ~ ) f
~
.....
- - .
o c c u p i e d
.
.
.
137
.
.
.
.
.
.
.
.
.
.
orbitals . . . .
::~::,y~/j///y/'//,//////~
.
.
-
|
.
.
.......................
.
.
.
-
.
.
.
.
-
.
-
-
-
. //Y/,
Fig. 6. Schematicenergy-leveldiagram of CoO. Co 3d orbitals are actually not completelyisolated becauseof their small overlap. (1) Charge-transferinsulating gap betweenO 2p band and Co first unoccupied 3d orbital. (2) The minimum energy for Co 3d --*O 3s conduction band. (3) The Hubbard U of Co 3d electron. (4) The small energy gap betweentwo localizedcobalt 3d orbitals. Our explanation of the insulating nature of CoO is as follows. The overlap of the excited 3d electrons is too small to form a metallic band, but it is sufficient for the hole to migrate through the crystal by electron exchange. The insulating nature of CoO is determined by the competitive result of several excitations. From the results of our calculations, CoO is a charge transfer insulator with a gap of about 5 eV (mostly from oxygen to cobalt). The calculated small energy gap should provide the activation energy of CoO. As compared to NiO, fewer experiments have been carried out for CoO. For the small energy gap, the direct evidence comes from the observed weak optical absorption lines below the threshold (done by Pratt and Coelho [30]), which show the dipole-forbidden Co occupied 3d to unoccupied 3d transitions. For the insulating gap, no experimental data on a cleaved single crystal CoO is available. Van Elp et al. [34] have performed a combined XPS, BIS study on Lio.olCoo.990. The measured band gap is 2.5 eV and it is found that the band gap is of an intermediate character, between Mott-Hubbard-like and charge-transfer-like.
The electronic structure of CoO is studied by a self-consistent cluster embedding calculation method. The picture of the ground state is as follows: A small energy gap separates the unoccupied and occupied cobalt 3d orbitals which are well localized. Below the cobalt 3d levels are two diffuse oxygen 2p bands, while above the 3d levels are empty oxygen 3s, cobalt 4s and oxygen 3p bands. The electron charge distribution properties of excited cobalt 3d states are studied theoretically. The excited 3d electrons are found to be well localized. An explanation for the insulating nature of CoO is proposed: the overlap of the excited 3d electrons and holes is too small to form a metallic band. So the insulating nature is determined by the competitive result of several excitations. The results of our calculations show that CoO is a charge-transfer insulator with the gaps of about 5 eV (mostly from oxygen ion to cobalt ion). The theoretical simulations of spin-disordered states with local AF order have produced a direct theoretical demonstration that the electronic structure in local AF pairs remains unchanged when CoO goes from paramagnetic phase to AF phase, which means that CoO remains an insulator in both phases. In conclusion, our calculated results lead to a natural interpretation of most experimental data. The results of CoO are consistent with that of NiO in a previous paper [1].
Acknowledgements I wish to thank Professor J. Callaway for helpful instructive advice.
References [1] H. Zheng, Phys. Rev. B 48 (1993-II) 14868. [2] R.J. Powell and W.E. Spicer, Phys. Rev. B 2 (1970) 2182. [3] Z.-X. Shen et al., Phys. Rev. Lett. 64 (1990) 2442. [4] Z.-X. Shen et al., Phys. Rev. B 44 (1990-II) 1817.
138
H. Zheng / Physica B 212 (1995) 125-138
[5] Z.-X. Shen et al., Phys. Rev. B 44 (1991-II) 3604. [6] T. Oguchi, K. Terakura and A.R. Williams, Phys. Rev. B 28 (1983) 6443. [7] T. Oguchi, K. Terakura and A.R. Williams, J. Appl. Phys. 55 (1984) 2318. [8] K. Terakura, A.R. Williams, T. Oguchi and J. Kubler, Phys. Rev. Lett. 52 (1984) 1830. [9] K. Terakura, T. Oguchi, A.R. Williams and J. Kubler, Phys. Rev. B 30 (1984) 4734. [10] D.E. Ellis, G.A. Benesh and E. Byrom, Phys. Rev. B 16 (1977) 3308. [11] H. Zheng, PhD Thesis, Louisiana State University (1993) (unpublished). [12] P. Hohenberg and W. Kohn, Phys. Rev. B 136 (1964) 864. [13] W. Kohn and L.J. Sham, Phys. Rev. A 140 (1965) 1133. [14] J.C. Stoddart and N.H. March, Ann. Phys. 64 (1971) 174. [15] M.M. Pant and A.K. Rajagopal, Solid State Commun. 10 (1972) 1157. [16] U. von Barth and L. Hedin, J. Phys. C 5 (1972) 1629. [17] A.K. Rajagopal and J. CaUaway, Phys. Rev. B 7 (1973) 1912. [18] R.W.G. Wyckoff, Crystal Structures, Vol. 1 (Interscience, New York, 1965). [19] C. Kittel, Introduction to Solid State Physics (Wiley, New York, 5th ed., 1976).
[20] A.K. Rajagopal, S. Singhal and J. Kimball (unpublished) as quoted by A.K. Rajagopal, in: Advances in Chemical Physics, Vol. 41, eds. G.I. Prigogine and S.A. Rice (Wiley, New York, 1979) p. 59. [21] A.J.H. Wachters, J. Chem. Phys. 52 (1970) 1033. [22] F.B. van Duijneveldt, IBM Research Report No. RJ945 (1971) (unpublished), [23] S. Huzinaga, Gaussian Basis Sets for Molecular Calculations (Elsevier, New York, 1984). [24] C.G. Shull, W.A. Strauser and E.O. Wollan, Phys. Rev. 83 (1951) 333. [25] D.C. Khan and R.A. Erickson, Phys. Rev. B 1 (1970) 2243. [26] W.L. Roth, Phys. Rev. 110 (1958) 1333. [27] D. Herrmann-Ronzaud, P. Burlet and J. Rossat-Mignod, J. Phys. C 11 (1978) 2123. [28] Shin Imada and Takeo Jo, J. Magn. Magn. Mater. 104-107 (1992) 2001. [29] Helge Johansen and Ulf Wahlgren, Mol. Phys. 33 (1977) 651. [30] G.W. Pratt and Roland Coelho, Phys. Rev. 116 (1959) 281. [31] N.C. Tombs and H.P. Rooksby, Nature 165 (1950) 442. [32] G.J.M. Janssen and W.C. Nieuwpoort, Phys. Rev. B 38 (1988) 3449. [33] Richard L. Martin, J. Chem. Phys. 98 (1993) 8691. [34] J. van Elp et al., Phys. Rev. B 44 (1991-I1) 6090.
Physica B 212 (1995) 125-138
Electronic structure of CoO Haoping
Zheng 1
Department of Physics and Astronomy, Louisiana State University. Baton Rouge, LA 70803, USA Received 28 July 1994; revised 24 October 1994
Abstract
The electronic structure of CoO is studied by a self-consistent cluster-embedding method. The localized and band properties are found in both ground state and excited states. An explanation for the insulating nature of CoO is proposed which is consistent with the one for NiO proposed previously. The simulations of both the spin-ordered states and the spin-disordered states (with local antiferromagnetic order) give a good description of the magnetic properties of CoO. It is found theoretically that CoO remains as an insulator when it goes from the paramagnetic phase to antiferromagnetic phase.
1. Introduction For more than five decades, the insulating nature of the transition-metal monoxides has been a continuing problem.. Due to historical reasons, NiO and CoO are considered as the representative transition-metal monoxides. In our previous paper [1], we reported the calculated electronic structure of NiO by using a self-consistent cluster-embedding (SCCE) method. The concept of double energy 9aps has been proposed to explain the insulating nature of NiO, which finds much experimental support. In this paper, the electronic structure calculated for CoO is presented. We show that the insulating nature of CoO can also be explained by the concept of double energy gaps. Experimental results show that NiO and CoO have similar properties. Both have the sodium chloride structure. Both show an antiferromagnetic transition at temperatures of 525 and 289 K, respectively. NiO and CoO are excellent insulators in
~Present address: Physics Department, Tongji University, Shanghai 200092, China.
both paramagnetic and antiferromagnetic phases. The optical-absorption coefficient of CoO is similar to that of NiO [2]. Similar photoemission spectra have been observed for NiO and CoO [3-5] However, compared with NiO, the situation in CoO is much less clear. Spin-polarized energy band calculations have obtained a small energy gap for NiO [6-9]. But CoO has an odd number of electrons per unit cell, which makes it more difficult for the band theory to reconcile its insulating nature. Even using the Slater approach to antiferromagnetism (each unit cell contains two CoO molecular units so that the number of electrons per cell is even), the results of spin-polarized energy band calculations still show a metallic nature for CoO in both paramagnetic and antiferromagnetic phases [6,9]. The self-consistent cluster embedding (SCCE) method was originally proposed by Ellis et al. [10] in 1977. Recently, we have proved that under certain conditions, the SCCE method separates the cluster-electrons from the background-electrons without changing the total energy of the system [ 1]. We call this electron ehar#e renormalization, which preserves the localized and band properties of the electrons in an embedded cluster. In addition, we
0921-4526/95/$09.50 (~i 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 1 100-1
H. Zheng / Physica B 212 (1995) 125-138
126
have developed a procedure to simulate the magnetic structure by using real atoms and ions, which makes this method useful in studying the magnetic properties of both spin-ordered and spin-disordered states [11]. As in the case of NiO, the calculation on CoO shows success: the properties of CoO can be interpreted naturally. The organization for the rest of this paper is as follows. Section 2 describes our theoretical model. Section 3 explains our computational procedures. The electronic structure of CoO is described in Section 4. Section 5 summarizes the results.
In a crystal, p2(r) can be obtained by using the crystal periodicity. Here we have introduced the orthogonality constraint, Vor, which is applied to the cluster-electrons in the surrounding core areas: M2
For :
Zi 2~j=l I r - e j l
if r is in the core of surrounding atoms,
0
otherwise.
(5)
During the calculation, Vor keeps the Pl out of the core area of the surrounding atoms. So within certain precision, we have
Pl (r) Vor(r) = 0
(6)
2. Theoretical model
A brief description of our theoretical model of the SCCE method is as follows. For a cluster of MI atoms (charge density Pl) embedded into an atomic array of M2 atoms (charge density P2), the total ground-state energy of the system (MI + M2 = M atoms) can be written [,12, 13] as
E~[p] = T[p] + EIc[P] + jfj f p(r)p(r') ~Z~7~ drdr' M
- 2 ~
~
p(r)Zi •
. ~
,--ur-e
~
i=1 J Ir - Ri[
ij
ZiZj
I~-RjI'
(1)
id:j
where p = p~ + P2. T[,p] is the kinetic energy of the noninteracting electrons, and Exc I-p] is the exchange-correlation energy. Ri and Zi are the position and charge of i'th nucleus, respectively. Atomic units are used here (e 2 = 2, h = 1, 2me = 1). Suppose we already know P2, by using the forms Pl (r) = p~P(r) + p~"(r) = ~
~
[~bf (r)l 2,
(2)
o occupied I
T[,pl] -- ~
~,
fq~'(r)(-V2)c~(r)dr,
(3)
occupied I
we derive the basic formula of the SCCE method [1]: { _ VZ + 2 fpl(r')_+ p z ( r ' ) d r ' - 2
J
Ir-r'l
~ Z, i=1 I r - e i l
+ V ~c(r) + Vo~ ~P7(r) = 27 ~b7(r). J
(4)
everywhere. This means that, although Eq. (4) contains an artificial potential Vor, if the condition (6) is satisfied with high precision, begin from Eq. (4), we can still derive the Kohn-Sham ground-state energy functional (1) which contains no Vor. Thus the separation of the cluster-electrons from the background-electrons does not change the total energy of the system. According to extended Hohenberg-Kohn theorem [14-17], unchanged total ground-state energy may mean unchanged total electron charge density. So the electron charge density calculated from Eq. (4) should be close to the real one and the embedded cluster can represent a portion of a solid. We call this electron charge
renormalization. In a solid, a band electron (here we mean it is shared by all atoms in the crystal) is usually described by Bloch plane waves. It is difficult to describe such a band electron directly by a finite cluster. What we can do is to try make an electron charge renormalization, which restricts the electrons around the cluster under the condition that the total ground-state energy and the total electron charge density are unchanged. Then it is reasonable to expect that both localized and band properties of the electrons are preserved in an embedded cluster, which means a band electron is now shared only by the atoms in the cluster, and a localized electron is still localized to the same atom. The total ground-state energy Ec is proportional to the number of cells. We have introduced a formula to represent the energy of an embedded
O1T !CO2
H. Zheng / Physica B 212 (1995) 125-138
cluster [1]:
E1 [N1, M,] = T[p,] + Ex¢, [p,,p2] +
i ~Pl(r)pl(r') a a ' f fP' (r)p21rt) j -~-_-r7] urur + j j ir--r-;I drdr' M
I
127
[ Pl (r)Zi d r , 2/='~ 1~ J [r -- Ril
COIL.) (7)
where EI~I [Pl, P2] = ~ P~ ex~(pup, pdn)dr. Fixed nuclei were assumed and the Coulomb interaction energy between the nuclei was removed. Definition (7) makes this method useful in the study of magnetic structure.
3. Computational procedure A general program package for an embedded cluster system using Gaussian orbitals has been developed by the author. Eq. (4) is solved self-consistently. The computational procedure, geometrical arrangement and embedding procedure used here are the same as that of the embedded Ni202 cluster [1]. Only a brief description is given here. Fig. 1 shows the four-atom Co202 cluster with the lattice constant 4.2667 ,A, the value of the lattice constant in crystalline CoO [18]. The number of surrounding atoms is 124. The number of surrounding point charges is 5380, with + 21el charge in Co sites and -21el charge in O sites. On the outmost boundary, fractional (½, ¼, and ~) charges are put on the faces, edges, and corners. This makes the Madelung constant at the positions of cluster atoms equal to 1.747565, which is the same as the exact Madelung constant of an ionic crystal with the sodium chloride structure [19]. The appropriateness of this embedding arrangement has been tested in the calculation of NiO. We use the von Barth and Hedin [16] form of the exchange-correlation potential, as parameterized by Rajagopal and co-workers [20]. A total of 240096 points is used in the Vxc calculation. In order to solve Eq. (4), the single-particle wave functions are expanded into a set of Gaussian orbitals. Table 1 gives basis set for CoO which was originally provided by Wachters [21] and van Duijneveldt [22]. A d polarization function is in-
02
Fig. 1. Four-atom Co202 cluster. cluded in the basis of oxygen [23]. Compared with the original bases, eight exponents have been changed, four exponents have been inserted and 19 diffuse exponents have been added. The ratios between changed and added adjoining exponents are about 2. A test of the basis set in two-atom-cluster calculation showed satisfactory convergence. The orthogonality constraint Vor of Eq. (5) is evaluated numerically in each of the surrounding atoms and point charges. The atomic core cutoff radius ro is adjusted according to two simple rules: (1) avoid numerical instability; (2) keep the total energy minimum. The results are 0.387 a.u. for cobalt atom and 1.132 a.u. for oxygen atom. In the calculation of Vor, 1376 points and 2494 points are used for the cobalt and oxygen cores, respectively. A two-step method is used because of the lack of disk-storage space. In the first step, a two atom Co-O embedded cluster is calculated with the selfconsistent determination of the surrounding charge density. In the second step, a four-atom Co202 embedded cluster is calculated with fixed surrounding charge density P2. Initially, Pe is built from Pl of the first step. Then t92 is built from p~ of the new results of four-atom Co202 cluster again and again. When the differences of all eigenvalues are less than 0.0002 Ry, and the difference of the total energy is less than 0.0007 Ry, the iterations were stopped. All background charge densities, P2, are built from the same Pl and are fixed during the calculation. The SCF iteration is only with respect to p~.
4. Results for an embedded Co202 cluster 4.1. Cluster ground state We build P2 from Pl according to the antiferromagnetic structure [24]. After 60 SCF iterations, the
H. Zheng / Physica B 212 (1995) 125-138
128 Table 1 Original
Basis set of cobalt atom s(11) Same Same Same Same Same Same Same Same Same Same Same Same 0.122266 0.044172
d(12) Same Same Same 1.4433 0.449965
Basis set of oxygen atom S(II) Same Same Same Same Same Same Same Same Same Same Same Same Same
d(1) Same
O u r basis
Original
Exponent
Coefficient
270991.0 39734.8 9057.46 2598.21 868.200 323.431
3.100D - 04 2.420D - 03 1.238D-02 4.849D-02 1.467D-01 3 . 2 1 3 D - 01
Our basis Exponent
Coefficient
1636.21 390.903 127.884 49.2413 20.7512
2.9600D 2.3360D 1.0343D 2.7954D 4.3268D
p(lO)
130.860 56.1219
4.0497D - 01 1.9961D - 01
18.9219 7.95238 2.19754 0.846713 0.205000 0.129199 0.060093 0.027950 0.013000
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
44.9774 12.5690 4.24422 2.20180 1.142242 0.592568
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
105374.95 15679.240 3534.5447 987.36516 315.97875 111.65428 42.699451
1 . 4 3 0 0 D - 04 1.1230D - 03 5 . 9 8 0 0 D - 03 2 . 5 5 6 2 D - 02 9 . 2 5 9 0 D - 02 2.81749D-01 6.77164D-01
17.395596 7.4383090 3.2228620 1.2538770 0.49515500 0.19166500 0.09043500 0.04267100 0.02013400 0.00950000
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
1.154000
1.0000
Same Same Same Same Same Same Same Same Same
0.1219
p(13) Same Same Same Same 2.102525 0.850223 Same 0.128892
9.20368 3.81779 1.58762 0.624660 0.170440 0.083140 0.040550 0.019780 0.009650
1.0000 1.13000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
0.307410 0.159477 0.082733 0.042920 0.022266 0.011551
1.0000 1.0000 1.0000 1,00130 1.0000 1.0000
200.00000 46.533367 14.621809 5.3130640 2.6675250 1.3392810 0.6724120 0.3375970 0.1713690 0.0869890 0.0441570 0.0224150 0.0113780
1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
-
03 02 01 01 01
H. Zheng / Physica B 212 (1995) 125 138
total energy has converged within 10-7 Ry. There are 0.00095 cluster-electrons remaining in the surrounding core regions, which means the condition (6) is well satisfied. Table 2 gives the eigenvalues and the Mulliken populations of spin-up orbitals. The spin-down orbitals are the same as the spin-up orbitals, except for the exchange of atoms 1 and 2. The Mulliken population analysis shows that each cobalt 3d orbital is mostly attached to a particular cobalt ion, while each oxygen 2p orbital is shared by two oxygen ions. Here we introduce the concept of charge overlap between two orbitals i and j: (Charge overlap)/j = f Minimum (pi(r), pj(r)) dr. (8) Unless otherwise specified, the "overlap" in this paper means charge overlap. The overlap between the two orbitals # 35 (spin-up and spin-down, both are cobalt 3d electrons but attached to different cobalt atoms) is 3.16%. In contrast, the overlap between the two orbitals # 28 (oxygen 2p electrons) is 84.97%. The overlap between the two orbitals # 39 (oxygen 3s electrons) is 92.52%. The picture of the ground state is the coexistence of localized and band properties: a 0.13 eV energy gap separates the unoccupied and occupied cobalt 3d orbitals which are well localized (this energy gap is too small to make CoO an insulator and will be discussed in Section 4.4). Each 3d orbital is mostly attached to a particular cobalt ion. Below the 3d levels are two diffuse oxygen 2p bands; above the 3d levels are empty oxygen 3s, cobalt 4s and oxygen 3p bands. The Mulliken population analysis show that the spin magnetic moment of cobalt ion is 2.94~B. The experimental data for the total magnetic moment of cobalt ion in CoO (including orbital contribution) is 3.35/~B [25], 3.8/~B. [26] and 3.8/~B [27]. The X-ray absorption spectroscopy shows that for Co 2 + ion the contribution of orbital-moment and spin are about 0.5-0.8/~ and 2.8/~B, respectively [28]. The Co202 clusters is in an antiferromagnetic state. Two cobalt ions have opposite spin magnetic moments, and two oxygen ions have no spin magnetic moment, so the total magnetic moment of the cluster is zero. This is the result of 90 ° super ex-
change effect.
129
4.2. Optical properties Fig. 2(a) shows the normal-emission angle-resolved photoemission experimental data and the results of energy-band calculations of CoO reported by Shen et al. [4]. Fig. 2(b) shows our calculated eigenvalues. For the reasons discussed in the previous paper [1], we simply use the eigenvalues in Table 2 as the ionization potential. Unfortunately, no off-normal-emission experimental data are available. For NiO, the off-normal-emission experimental data show more features than the normal-emission data [5]. Some features may be missing in Fig. 2(a) because of the rigorous selection rules in a normal-emission experiment. In general, our results are similar to the energy-band calculation. Both show that cobalt 3d bands are above the oxygen 2p bands, which is in agreement with the experimental data. The angle-integrated photoemission data of CoO show a satellite located about 8.5 eV lower than the first peak [4]. Shen et al. assigned it as a d 6 satellite and use it to derive a 9-11 eV Ud value. In our calculation, no such satellite appears. As Co 3d electrons are well localized and strongly correlated, we think the satellite may come from the ionization of Co 3+ ion following the ionization of Co 2 + ion (double holes). No confirmation has been made since we are unable to calculate a full relaxed ionization state. Our high-quality basis set and relatively accurate embedding procedure make the lowest several unoccupied orbitals acceptable, which enables us to interpret the experimental optical-absorption data. Fig. 3(a) shows the experimental result for the optical-absorption coefficient of CoO obtained by Powell and Spicer [-2]. Fig. 3(b) shows the optically allowed transition energies obtained from the differences of the calculated eigenvalues. Following the discussion of Ref. [1], the real value will be a little larger than that in Fig. 3(b) if the transition involves a large electron rearrangement. Included are all optically allowed transitions, from cobalt 3d and oxygen 2p valence orbitals to the unoccupied cobalt 3d, oxygen 3s, cobalt 4s and oxygen 3p orbitals. The highest unoccupied orbital counted is an oxygen 3p orbital with eigenvalue of 0.5598 Ry (7.6 eV). Part A of Fig. 3(b) contains the energies of the charge transfer transitions from oxygen 2p orbitals to the unoccupied cobalt 3d orbitals. Because the
H. Zheng / Physica B 212 (1995) 125 138
130
Table 2 E i g e n v a l u e s a n d Mulliken p o p u l a t i o n s of C o 2 0 2 cluster in g r o u n d state (spin-up orbitals) Orbital
No.
O 3p
44
0.5598
0.01
- 0.01
Co4p
43
0.5175
0.50
- 0.04
0.01 - 0.04
Co4s
42
0.4161
0.78
0.03
- 0.03
-
0.3852 0.2290
1.01
- 0.25 - 0.29
- 0.04 - 0.06
- 0.04
O 3s
41 40
- 0.06
1 . 2 5
39
0.1256
0.05
- 0.15
- 0.02
- 0.02
1.13
38 37
- 0.2014 - 0.2543
0.02
36
- 0.2830
C o 3d
E n e r g y (Ry)
Co s
Co p
Co 1 d
Co 2 d
O 1p
O 2p
0.50
0.50
0.29
0.29
0.23 0.16
0.23 0.16
0.08
0.08
0.01
0.01
0.98
0.01
0.01
0.97 0.02
0.01 0.01 0.04
0.01 0.01 0.04
0.01
0.01
0.01 0.03
0.01
O s
- 0.23 -
0.95 0.98
-
0.01
0.02 0.01
Note: A b o v e are the u n o c c u p i e d orbitals C o 3d
O2p
O 2s C o 3p
C o 3s
35(Ef) 34
- 0.2926 - 0.3277
33
- 0.3959
32 31
- 0.4525 - 0.4766
0.01
30 29
- 0.4873 - 0.5111
0.02
28 27 26
Co2p
C o 2s C o ls
0.01
0.97 0.01
0.89 0.97 0.94
0.01
0.94 0.84
0.01 0.01
- 0.6190
0.16
- 0.6350 - 0.6906
0.06
0.02 0.01
0.05
25
- 0.7069
0.01
0.08
24
- 0.7103
0.17
- 0.07
0.04
23
- 0.7612
0.14
0.05
0.01
22 21 20 19
-
1.6875 1.6952 4.3645 4.4139
- 0.05
- 0.03 - 0.03 1.00 1.00
18
- 4.4364
1.00
17
- 4.5901
1.00
16
- 4.6347
1.00
15 14
- 4.6570 - 6.9198
1.00
- 7.1479
1.00
13 O ls
0.01
12 11 10 9 8 7
-
6 5 4 3 2 1
-
0.03 0.02 0.08 0.47
0.47
0.45
0.45
0.06
0.43 0.43 0.44
0.43 0.43 0.44
0.01 0.01
0.01 0.01
- 0.08 1 . 0 3
1.07
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
0.41
0.02 0.08 0.41
0.03
1.00
37.2358 37.2358 55.9840 55.9892 55.9913 56.0769
56.0816 56.0837 64.3784 64.5002 - 551.0222 - 551.0232
0.02
-
H. Zheng / Physica B 212 (1995) 125-138
Co 0 3d 2p
'
i
'
I
'
I
'
C o O (001)
~
I
~
vv
t6
'
CoO ABSORPTION
.~ 14
-
131
-
COEFFICIENT
E
0 ~J
12
I0 PI36
>•
~08
-4
g •
~5 E ~
PRO
&5
0
~o4 -6 O~
o
p 2
i 4
6
(a) o
o
~
c
v
A
I 0.2
,
(b)
1
[ 0.4
,
I 0.6
C~slal Momentum
(b)
, I0 12 14 16 IR PHOTON ENERGY {eV}
~ ~
0 ,
8
,
I
,
0.8
B
X
cobalt 3d orbitals are well localized and oxygen 2p orbitals are diffuse, transfering one oxygen 2p electron to an unoccupied cobalt 3d orbital will cause large electron rearrangement. The real transition energies should be a little larger than that in part A. Considering this factor, part A is reasonably in agreement with the experimental data. The smallest transition energy in part A is 4.75 eV. We estimate that the main optical absorption edge is about 5 eV. Parts B, C, and D in Fig. 3(b) contain the transition energies of O 2p ~ O 3s, O 2p ~ Co 4s, and Co 3d ~ O 3p (Co 4p), respectively. In general, they are lower by about 2 eV than the experimental data. We think the error comes mainly from the inaccuracy of the unoccupied O 3s, Co 4s and O 3p (Co 4p) orbitals. In cluster calculations, usually, the higher the unoccupied orbital is, the less accurate is its energy. Although they have the correct order (by which we mean they are 'acceptable'), the orbitals of O 3s, Co 4s and O 3p (Co 4p) are certainly less accurate than the unoccupied Co 3d orbitals. Considering this factor, we interpret that the peak
22
24
26
c
III I1 Illl [nml D
(a)
Fig. 2. (a) The photoemission experimental data and the results of energy-band calculations of CoO [4]. (b) The eignevalues of embedded Co~O2 cluster. The position of the first Co 3d orbital in (b) is aligned to the first experimental data in (a) (represented by V).
, 20
Fig. 3. (a) The experimental results of optical-absorption coefficient of CoO [2]. (b) The optically allowed transition energies obtained by using the eigenvalues.
around 13.6 eV comes from part B; the peak at 18.4 eV comes from part C and part D (part D involves a large electron rearrangement so the real energies should be even larger). The weak optical-absorption peaks below the threshold are the 3d-3d dipole-forbidden transitions of the same cobalt ion. Johansen and Wahlgren [29] have calculated these d-d excitation spectra by using SCF/CI techniques on a CoO6 10 cluster. In our calculation, we can get twelve 3d 3d transition groups by using the eigenvalues in Table 2. Their average values are (in eV) 0.13, 0.57, 1.00, 1.24, 1.63, 1.93, 2.31, 2.69, 3.10, 3.46, 3.82 and 4.21. The experimental data between 0.7 and 2.9 eV are about 0.83, 0.90, 1.03, 1.24, 1.63, 2.03, 2.30, 2.49 and 2.61 eV [30]. Finally, by using the eigenvalues, the cobalt 3p to 3d threshold is 56eV, close to the experimental result of about 60 eV.
4.3. Magnetic properties This subsection describes the simulation of spin disordered states with local antiferromagnetic (AF) order of CoO.
H. Zheng / Physica B 212 (1995) 125-138
132
First, we estimate the stability of the local antiferromagnetic (AF) order. We calculated the total energy of the ferromagnetic state. The result is 3.51 eV (41000 K) above the energy of the AF state. This indicates very stable local AF order. So it is reasonable to assume that in the paramagnetic state, local AF order still remains and the local moments of cobalt ions persist essentially unchanged. In order to simulate the spin disordered states with local AF order, two geometrical restrictions have been applied: (1) the total number of spin-up cobalt ions should equal the total number of spin-down ions; (2) in any cube (containing four cobalt ions and four oxygens ions), the four cobalt ions should not have the same spin direction. A random-number-generating subroutine is used to choose the spin-up and spin-down atoms, then the two geometrical restrictions mentioned above are applied to the 18 nearest Co ions (the change of spin direction of the other Co ions will not change the result).
Table 3 gives our results. The last two rows show the special cases which violate our restrictions. One row has all spins up, the other has each clustercobalt ion surrounded by spin-parallel cobalt ions. Both cases cannot happen in practice because of the local AF order. Apparently, the AF structure has the lowest energy among the disordered structures. Because of the local AF order, the disordered states are actually not completely disordered, so the change of the system entropy may be small. If we neglect the change of system entropy, by averaging the eight disordered results in Table 3, we get the N6el temperature of 1162 K; the experimental result is 289 K [31]. We think the discrepancy comes mainly from the imperfection of the exchange-correlation potential we used. Table 3 also shows the energy gaps, the optical absorption edges and the spin magnetic moment of each cobalt ion. It is clear that they are independent of the long-range spin order. The calculated spin
Table 3 Results of a four-atom CO202 cluster with different atomic spin direction arrangement of surrounding atoms (the Co202 cluster is in antiferromagnetic state) No.
Atomic spin direction arrangement
18 nearest Co atoms up
1 2 3 4 5 6 7 8 9 10
Antiferromagnetic (31 up, 31 down) Disordered 1 (31 up, 31 down) Disordered 2 (31 up, 31 down) Disordered 3 (31 up, 31 down) Disordered 4 (30 up, 32 down) Disordered 5 (31 up, 31 down) Disordered 6 (31 up, 31 down) Disordered 7 (31 up, 31 down) Disordered 8 (31 up, 31 down)
All spins up
Total energy
Energy gap
Absor, edge
Spin magnetic moment (/~B)
(K °)
(eV)
(eV)
Co 1-up
Co 2-down
down
9
9
0
0.130
4.572
2.9397
2.9397
9
9
+ 1202
0.102
4.545
2.9355
2.9379
9
9
+ 1587
0.100
4.539
2.9370
2.9375
9
9
+ 704
0.128
4.570
2.9393
2.9380
9
9
+ 711
0.119
4.559
2.9399
2.9377
9
9
+ 1444
0.106
4.562
2.9371
2.9393
9
9
+ 1802
0.100
4.547
2.9373
2.9371
9
9
+ 399
0.120
4.560
2.9402
2.939
9
9
+ 1446
0.103
4.546
2.9364
2.9379
18
0
+ 1383
0.149
4.599
2.9366
2.9408
9
9
- 1173
0.152
4.590
2.9412
2.9412
(62 up, 0 down) 11
Special case (31 up, 31 down)
H. Zheng/Physica B 212 (1995) 125 138
magnetic moments of cobalt ions in spin disordered states are almost the same as that in the AF state, which shows the consistency of our assumption. Table 4 shows the eigenvalues and the Mulliken populations of one of the spin-disordered states (disordered 1 in Table 3). Tables 4 and 2 are almost identical, which means they have similar electronic structure. Actually, we find that the electronic structure of the cluster remains almost the same in all spin-disordered states as in the AF state. This means that in a local AF pair, the orbitals are nearly the same for both paramagnetic and AF phases. The photoemission experimental data support our results: the photoemission spectra taken in the two phases of CoO are identical [4]. From the discussion of the next section, we can say that CoO remains as an insulator when it transforms from the AF to paramagnetic phase. This result is in agreement with the observed properties of solid CoO.
4.4. Energy gap In this section, the same idea and method as in the case of NiO [1] are used to give an explanation of the insulating nature of CoO. It is clear that our energy gap of 0.13 eV cannot make CoO an insulator. From Table 5(a), we know the overlap of the highest occupied 3d orbital (orbital 35) is 3.16%, the overlap of the first unoccupied 3d orbital (orbital 36) is 1.97%. Using the idea and explanation of our previous paper [1], it is clear that these two orbitals cannot form metallic bands, so the ground state of cluster Co202 is an insulating state. Fig. 4(a) shows the charge distribution of the ground state along the line connecting Co 1 and O 1 ion. The charge distribution of the ground state along the line connecting two cobalt ions is shown in Fig. 4(b). It is clear that the electrons of cobalt ions are well localized and there is almost no overlap between the two cobalt ions. In order to discuss the insulating gap, the Hubbard U parameter for cobalt 3d electron is estimated first. For the embedded Co202 cluster with fixed P2, the ionization potential of the orbital 35 and the electron affinity of the orbital 36 (see Table 2) were calculated by using the transitionstate method. We take the point of view that
133
U should be the difference of ionization potential and the electron affinity. The results is Udd ~ l l.0eV. Because of limited electron rearrangement during the transition-state calculation (p2 fixed), the real Udd value should be smaller than that one. Janssen and Nieuwpoort [-32] and Martin [33] have reported the long-range polarization contribution which reduces the calculated Udd of a cluster dramatically. Unfortunately, we are unable to calculate the Udd with full relaxation. The best estimate we can make is that the Udd is probably in the range of 7-9 eV. The electronic structures of 13 excited states have been calculated. The results of all six 3d-3d transitions involving the same cobalt ion and the same spin direction (spin-preserving transitions) are presented in Table 5(a). These results are reliable because the electron rearrangement is very small during the transitions (compared with the ground state in Table 5(a), the changes of the eigenvalues are less than 0.015 Ry). In Table 5(a), the maximum overlap of an excited 3d electron is 2.750, the maximum overlap ofa 3d hole is 3.25%. Fig. 5 shows the charge distribution of state 2 along the line connecting two cobalt ions. Table 5(b) gives the results of six 3d 3d spin-flipped transitions between the orbitals of the same cobalt ion. The probabilities of these transitions are much lower than that in Table 5(a) because they are dipoleforbidden and they reduce the spin magnetic moment of cobalt ion to about 1/~B. The electron rearrangement in these transitions are larger than that in Table 5(a) due to the change of spin magnetic moment and the change of exchange-correlation energy. Nevertheless, we think it is acceptable to get the localized and band information from these results. Our experience shows that the calculations of spin-flipped transitions are convergent except the one from orbital 32 to 38. The transitions from the lowest occupied 3d orbital (orbital 29) to the three unoccupied 3d orbitals are calculated. Four of the five transitions from the occupied 3d orbitals to the highest unoccupied 3d orbital (orbital 38) are also calculated. We think it is sufficient for our argument, because the excited electron in orbital 38 and the hole in orbital 29 or 30 usually have the maximum overlap. In Table 5(b), the largest overlap of an excited 3d
H. Zheng / Physica B 212 (1995) 125-138
134
Table 4 E i g e n v a l u e s a n d M u l l i k e n p o p u l a t i o n s of C o 2 0 2 cluster in s p i n - d i s o r d e r e d 1 state (spin-up orbitals) Orbital
No.
O3p Co4p
44 43
0.5602 0.5170
Co4s
42
0.4169
0.80
O3s
41 40 39
0.3840 0.2291 0.1251
C o 3d
E n e r g y (Ry)
38
- 0.2030
37
-0.2559
36
- 0.2846
Co s
Co p
Co 1 d
Co 2 d
O s
O 1p
O 2 p
0.48 0.30
0.53 0.29
0.01
- 0.01
0.50
- 0.04
- 0.01 - 0.04
1.00
0.02 - 0.24
- 0.03 - 0.04
- 0.02 - 0.04
- 0.23 - 0.01
0.23 0.15
0.18
- 0.06 - 0.02
- 0.06 - 0.02
1.25 1.14
0.08
0.08
0.04
- 0.29 - 0.15
0.95
0.02
0.01
0.01
0.98
- 0.01
0.98
0.01
0.01
0.97
0.01 0.02
0.01 0.01
0.02
0.04
0.04
- 0.01
0.01 0.03 0.02
0.01 0.03 0.02
0.08 0.47
0.07 0.35
0.40
0.53
0.49 0.54
0.42 0.32
0.02
Note: A b o v e are the u n o c c u p i e d orbitals Co3d
O2p
O2s Co3p
Co3s O ls Co2p
Co2s C o ls
35(El)
- 0.2921
34 33
- 0.3271 - 0.3953
32
- 0.4516
31
- 0.4756
30
- 0.4873
29 28 27 26 25 24
-
0.5109 0.6186 0.6346 0.6896 0.7079 0.7104
23
- 0.7617
0.01
0.01
0.97 0.01 0.01 0.02
0.05
0.01
20 19 18
- 4.3671 - 4.4140 - 4.4363
1.00
17
- 4.5889
1.00
16 15 14
- 4.6346 - 4.6568 - 6.9206
1.00
- 56.0838 - 64.3788 - 64.5002 - 551.0226 -551.0233
0.01 0.02 0.01 0.02
0.14
- 1.6872
5 4 3 2 1
0.84 0.16 0.06 0.07 0.04
- 1.6949
-7.1474 37.2354 37.2364 55.9846 55.9895 55.9916 56.0768 56.0817
0.01
0.02 0.16
21
-
0.97 0.94 0.94
0.05 0.01 - 0.06
22
13 12 11 10 9 8 7 6
0.89
- 0.03 -0.05
- 0.03
0.01
0.03 0.06
0.27
0.58
- 0.08
0.46
0.43
1.03 1.07
0.01 0.01
0.01
1.00 1.00
1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1.00 1.00 1.00 1.00
H. Zheng/Physica B212 (199~ 125-138 ii-
10-
(a)
I0,.....,
9-
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9-
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01
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Fig. 5. The charge distribution of excited state 2 along the line connecting two cobalt ions. 10
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Fig. 4. The charge distribution of ground state 0fC0202 cluster: (a) along the line connecting Co 1 and O 1 ions; (b) along the line connecting two cobalt ions.
electron is 4.63%, the largest overlap of a 3d hole is 10.73%. The result of state (8) in Table 5(a) is not reliable because its large electron rearrangement is partially restricted by the fixed P2. The only certain conclusion we can draw from it is that each 2p hole
is shared by two oxygen ions. It is interesting to find that orbital 36 is still well localized (which means Co ÷ is localized), although we cannot confirm it absolutely. In conclusion, similar to the ground state, the excited electrons and holes have also both localized and band properties. For all spin-preserving 3d-3d transitions, the overlaps of excited orbitals are less than 2.8%, and the overlaps of holes are less than 3.3%. For spin-flipped 3d-3d transitions, the overlaps of excited orbitals may be less than 4.7%; the overlaps of holes may be about 11%. The holes in oxygen 2p orbitals are diffuse, which means the holes can move freely in solid. CoO has the following four properties (see Fig. 6): (1) the minimum energy required to transfer an electron from oxygen 2p bands to a cobalt site is about 5 eV (the main optical absorption edge); (2) the minimum energy required to transfer a cobalt 3d electron to the oxygen 3s conduction band is about 6 eV; (3) the minimum energy required to transfer an electron from one cobalt ion to another cobalt ion far away is about 8 eV (the Hubbard U parameter for cobalt 3d electron); (4) the 0.13 eV energy gap is between two localized cobalt 3d orbitals. The 3d excited orbitals and holes of cobalt ion are well localized.
H. Zheng / Physica B 212 (1995) 125 138
136
Table 5(a) Eigenvalues and Mulliken populations of C o 2 0 2 cluster in the g r o u n d state and seven spin-preserving excited states (spin-up orbitals) State
Orbital No.
Occupation number
Eigenvalue (Ry)
Co 1 (%)
Co 2 (%)
O 1 (%)
O 2 (%)
Overlap (%)
38 37 36 35 (Ef) 34 33 32 31 30 29 28
0 0 0 1 1 1 1 1 1 1 1
- 0.2014 - 0.2543 - 0.2830 - 0.2926 - 0.3277 - 0.3959 - 0.4525 - 0.4766 - 0.4873 -0.5111 - 0.6190
0.4 0.7 0.0 0.5 0.1 89.6 97.8 93.5 95.0 84.1 15.6
94.9 99.4 98.2 97.2 97.0 0.5 1.0 0.1 0.6 0.9 1.8
2.3 0.0 0.9 1.1 1.4 4.9 0.5 3.2 2.2 7.5 41.3
2.3 0.0 0.9 1.1 1.4 4.9 0.5 3.2 2.2 7.5 41.3
2.65 1.12 1.97 3.16 3.17 5.20 2.37 6.76 5.77 16.36 84.97
(2) 35 to 36
36(exc.) 35(hole)
1 0
- 0.2860 - 0.2944
0.0 0.5
98.2 97.3
0.9 1.0
0.9 1.0
1.95 3.05
(3) 35 to 37
37(exc.) 35(hole)
1 0
- 0.2516 -0.2893
0.7 0.5
99.6 97.1
- 0.1 1.2
- 0.1 1.2
1.05 3.16
(4) 35 to 38
38(exc.) 35(hole)
1 0
- 0.2031 - 0.2931
0.4 0.5
94.7 97.2
2.4 1.1
2.4 1.1
2.75 3.25
(5) 34 to 36
36(exc.) 34(hole)
1 0
- 0.2734 - 0.3168
0.0 0.1
98.3 97.2
0.9 1.3
0.9 1.3
1.88 2.93
(6) 34 to 37
37(exc.) 34(hole)
1 0
- 0.2399 - 0.3130
0.7 0.1
99.6 97.3
- 0.1 1.3
- 0.1 1.3
1.04 2.88
(7) 34 to 38
38(exc.) 34(hole)
1 0
- 0.1894 - 0.3146
0.4 0.1
94.9 96.9
2.3 1.5
2.3 1.5
2.68 3.23
(8)
36(exc.) 35 34 28(hole)
1 1 1 0
0.0160 - 0.0076 - 0.0327 - 0.8205
0.2 - 6.9 0.0 0.3
97.7 69.9 98.6 0.2
1.0 18.5 0.7 49.7
1.0 18.5 0.7 49.7
2.02
(1) ground state
28 to 36
Table 5(b) Eigenvalues and Mulliken populations of
Co202 cluster
State
Orbital No.
Occupation number
(9) 29 to 36
36(exc.) 29(hole)
(10) 29 to 37
99.86
in six spin-flipped excited states (spin-up orbitals)
Eigenvalue (Ry)
Co 1 (%)
Co 2 (%)
O 1 (%)
O 2 (%)
Overlap (%)
1 0
- 0.3343 - 0.4410
0.1 93.8
97.8 2.2
1.1 2.0
1.1 2.0
2.50 8.41
37(exc.) 29(hole)
1 0
- 0.3002 - 0.4362
1.3 94.0
98.5 2.1
0.0 1.9
0.0 1.9
2.57 8.11
(11) 29 to 38
38(exc) 29(hole)
1 0
- 0.2506 - 0.4389
0.8 92.8
93.4 2.6
2.8 2.3
2.8 2.3
4.31 9.73
(12) 30 to 38
38(exc.) 30(1101e)
1 0
- 0.2695 - 0.4292
0.9 93.6
92.9 4.2
3.0 1.1
3.0 1.1
4.63 10.73
(13) 31 to 38
38(exc.) 31(hole)
1 0
- 0.2629 -0.4089
0.9 96.2
93.0 0.5
2.9 1.6
2.9 1.6
4.58 4.52
(14) 33 to 38
38(exc.) 33(hole)
1 0
- 0.2606 -0.3266
0.9 92.3
93.5 1.3
2.8 3.2
2.8 3.2
4.30 5.04
H. Zheng/Physica B 212 (1995) 125 138
5. Summary
I O 3s conduction band
2 (6ev) 3 (8ev)
uC°~l~!dpied ( f ; ! ~ ) f
~
.....
- - .
o c c u p i e d
.
.
.
137
.
.
.
.
.
.
.
.
.
.
orbitals . . . .
::~::,y~/j///y/'//,//////~
.
.
-
|
.
.
.......................
.
.
.
-
.
.
.
.
-
.
-
-
-
. //Y/,
Fig. 6. Schematicenergy-leveldiagram of CoO. Co 3d orbitals are actually not completelyisolated becauseof their small overlap. (1) Charge-transferinsulating gap betweenO 2p band and Co first unoccupied 3d orbital. (2) The minimum energy for Co 3d --*O 3s conduction band. (3) The Hubbard U of Co 3d electron. (4) The small energy gap betweentwo localizedcobalt 3d orbitals. Our explanation of the insulating nature of CoO is as follows. The overlap of the excited 3d electrons is too small to form a metallic band, but it is sufficient for the hole to migrate through the crystal by electron exchange. The insulating nature of CoO is determined by the competitive result of several excitations. From the results of our calculations, CoO is a charge transfer insulator with a gap of about 5 eV (mostly from oxygen to cobalt). The calculated small energy gap should provide the activation energy of CoO. As compared to NiO, fewer experiments have been carried out for CoO. For the small energy gap, the direct evidence comes from the observed weak optical absorption lines below the threshold (done by Pratt and Coelho [30]), which show the dipole-forbidden Co occupied 3d to unoccupied 3d transitions. For the insulating gap, no experimental data on a cleaved single crystal CoO is available. Van Elp et al. [34] have performed a combined XPS, BIS study on Lio.olCoo.990. The measured band gap is 2.5 eV and it is found that the band gap is of an intermediate character, between Mott-Hubbard-like and charge-transfer-like.
The electronic structure of CoO is studied by a self-consistent cluster embedding calculation method. The picture of the ground state is as follows: A small energy gap separates the unoccupied and occupied cobalt 3d orbitals which are well localized. Below the cobalt 3d levels are two diffuse oxygen 2p bands, while above the 3d levels are empty oxygen 3s, cobalt 4s and oxygen 3p bands. The electron charge distribution properties of excited cobalt 3d states are studied theoretically. The excited 3d electrons are found to be well localized. An explanation for the insulating nature of CoO is proposed: the overlap of the excited 3d electrons and holes is too small to form a metallic band. So the insulating nature is determined by the competitive result of several excitations. The results of our calculations show that CoO is a charge-transfer insulator with the gaps of about 5 eV (mostly from oxygen ion to cobalt ion). The theoretical simulations of spin-disordered states with local AF order have produced a direct theoretical demonstration that the electronic structure in local AF pairs remains unchanged when CoO goes from paramagnetic phase to AF phase, which means that CoO remains an insulator in both phases. In conclusion, our calculated results lead to a natural interpretation of most experimental data. The results of CoO are consistent with that of NiO in a previous paper [1].
Acknowledgements I wish to thank Professor J. Callaway for helpful instructive advice.
References [1] H. Zheng, Phys. Rev. B 48 (1993-II) 14868. [2] R.J. Powell and W.E. Spicer, Phys. Rev. B 2 (1970) 2182. [3] Z.-X. Shen et al., Phys. Rev. Lett. 64 (1990) 2442. [4] Z.-X. Shen et al., Phys. Rev. B 44 (1990-II) 1817.
138
H. Zheng / Physica B 212 (1995) 125-138
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