Ab initio cluster calculations of the magnetic properties of ZnO doped with transition metal ions

Chemical Physics 326 (2006) 297–307 www.elsevier.com/locate/chemphys

Ab initio cluster calculations of the magnetic properties of ZnO doped with transition metal ions Karin Fink

*

Lehrstuhl fu¨r Theoretische Chemie, Ruhr-Universita¨t Bochum, Universitatsstr. 150, 44780 Bochum, NRW, Germany Received 7 December 2005; accepted 15 February 2006 Available online 20 March 2006

Abstract The magnetic properties of transition metal (TM) doped ZnO (TM = Mn2+, Co2+, and Ni2+) were determined by an embedded cluster approach using ab initio methods. We performed CASSCF (complete active space self-consistent field) calculations on clusters with one and two transition metal centers, respectively. In particular, we were interested in the magnetic exchange coupling between TM centers which are directly connected by an oxygen bridge. Two Mn as well as two Co centers are weakly antiferromagnetically coupled. For the investigation on the magnetic properties of Ni2+ doped ZnO it was necessary to include spin orbit coupling in the calculations. While without inclusion of spin orbit coupling the ground state of an isolated Ni2+ ion in ZnO is a 3E state, which should show a first order Zeeman interaction with the magnetic field, Ni2+ has a non-magnetic A1 ground state, when spin orbit coupling is included. The results of our calculations are in good agreement with experimental data for low temperatures and low concentrations of the transition metal centers.  2006 Elsevier B.V. All rights reserved. Keywords: Magnetic exchange coupling; Diluted magnetic semiconductors; Ab initio calculations; Spin orbit coupling

1. Introduction Diluted magnetic semiconductors [1] are currently intensively studied because they are promising materials for spintronics. Since Dietl et al. [2] suggested that Mn doped ZnO should show room-temperature ferromagnetism caused by an exchange coupling between local spins and charge carriers as proposed by Zener [3], several research groups investigated TM (transition metal) doped ZnO in the last years. Reviews on this work have been published for example by Pearton et al. [4] or Prellier et al. [5]. However, the results of different groups do not agree at all. Spin glass behavior was observed by Fukumura et al. [6] and Rao and Deepak [7] for Mn doped ZnO, while Jung et al. found ferromagnetism with TC = 30–45 K [8] as did Sharma et al. with TC = 425 K [9]. Kittilstved et al.

*

Tel.: +49 234 3222125. E-mail address: [email protected].

0301-0104/$ - see front matter  2006 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2006.02.007

observed strong 300 K ferromagnetism for Mn doped ZnO which was additionally p doted [10]. The situation is similar for Co doped ZnO. Several authors report Curie temperatures higher than 280 K [11,12]. Norton et al. [13] suggested that the ferromagnetism could be caused by Co nanocrystallites embedded in ZnO. However, as for Mn doped ZnO, Rao et al. [7] did not observe ferromagnetism in Co doped ZnO. In a combined experimental and theoretical work Risbud et al. [14] proposed an antiferromagnetic coupling between nearest neighbors. For Ni doped ZnO, high temperature ferromagnetism was observed by Radovanovic and Gamelin [15]. These authors mention that the conversion from paramagnetism to ferromagnetism is strongly increased by increasing charge carrier concentration. At temperatures lower than 100 K they observed a strong decrease of the paramagnetism of isolated Ni2+ centers in ZnO which is attributed to the spin orbit coupling of the Ni2+ ground state in an environment with Td symmetry.

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Theoretical work on TM doped ZnO is usually based on supercell calculations using DFT methods. The first supercell calculations on magnetic exchange coupling constants have been performed by Towler et al. [16] on NiO and MnO. The exchange coupling is obtained from the energy difference between the high-spin state or ferromagnetic state and an Ising state in which the magnetic orbitals have alpha spin at one metal center and beta spin at the other. Moreira et al. [17] and Caballol et al. [18] have shown the mapping of these energy differences with the exchange coupling constant J and the similarity of this ansatz to the Noodleman approach for binuclear complexes [19,20]. The magnetic exchange coupling for transition metal centers in ZnO is obtained as the energy difference between the ferromagnetic and the spin glass state, which is equivalent to an Ising state [14,21–29]. For Mn doped ZnO without further n or p doping all calculations found the spin glass state to be most stable. For Co, Sato and Katayama-Yoshida [22–25] found a ferromagnetic ground state, while Spaladin et al. [28] obtained almost the same energies for the antiferromagnetic and the ferromagnetic phase. Sluiter et al. [21] performed a systematic study of the exchange interaction as function of the distance of the TM centers (nearest neighbors, next nearest neighbors, etc.). They showed that for Ni, Co, and Mn only the interactions with the nearest neighbors are important. We know from our previous work on transition metal compounds [30–33] that DFT calculations can only be used in cases, for which the ground state of the TM is spatially non-degenerate. Even for those cases, DFT results depend strongly on the chosen functional [34,35] and have to be handled with care. For spatially non-degenerate states, the exchange coupling of two local spins S1 and S2 can often be described by a Heisenberg operator H = 2JS1S2. In such cases, the splitting of the resulting spin states S = S1 + S2, S1 + S2  1, . . . , |S1  S2| is given by a Lande´ pattern EðS; S 1 ; S 2 Þ ¼ J ½SðS þ 1Þ  S 1 ðS 1 þ 1Þ  S 2 ðS 2 þ 1Þ.

ð1Þ

If the ground state is spatially degenerate, the superexchange coupling can no longer be described by a simple Lande´ splitting and spin orbit coupling (SOC) has to be considered [32,33]. This is certainly necessary for Ni doped ZnO, but so far, SOC was included in none of the calculations [21,25] and Ni doped ZnO was found to be ferromagnetic. In the present paper, we report on a series of wave function based ab initio calculations of the exchange coupling constant between TM centers in ZnO. Our investigation is restricted to the coupling between two nearest neighbor TM2+ ions because the exchange coupling decays rapidly with increasing distances between the magnetic ions. All our calculations are performed by means of an embedded cluster approach. Embedded cluster calculations [36–40] are widely and successfully used for local phenomena such as adsorption processes, d–d excitations in transition metal oxides, magnetic exchange coupling between two magnetic

centers, defects and so on. For calculations on the magnetic exchange coupling the embedded cluster approach allows for a detailed analysis of the different contributions to the magnetic coupling, such as superexchange and direct exchange. Spin orbit coupling was included in our calculations and its influence on the magnetic properties is investigated. 2. Embedded cluster model For small concentrations of transition metal ions the structure of the doped ZnO does not differ much from the bulk structure of pure ZnO. Prellier et al. [5] showed, that the c-axis lattice constant of Co doped ZnO changes in the concentration range of 0–25% linearly with the Co ˚ . Sluiter et al. [21] considered content, but only by 0.01 A relaxations of the atoms in the unit cell by fixed lattice constants in their periodic calculations and proved that the changes of the local structure in the surrounding of the transition metal centers are so small that neither the electronic structure nor the magnetic coupling are influenced. In their investigations on a Mn doped ZnO film, Wang et al. [29] observed a relaxation of the Mn–O distances of 4% for surface atoms of the film. In the second layer the Mn–O distance was already almost identical to the bulk Zn–O distance. Therefore, we do not expect large changes of the local geometrical structure by substituting bulk Zn ions by TM2+ centers. This is in agreement with our former work on F centers in ZnO [41]. As long as the charge of the center is not changed relaxation effects are small, even if the whole atom is removed. All our calculations were performed using the technique of embedded clusters. That means that we treat only a small cluster by quantum chemical ab initio methods, quantum cluster; this cluster is embedded in a large array of point charges (point charge field) which takes care of the correct electrostatic field for the atoms of the quantum cluster. The ligand field at the Zn ions is almost tetrahedral ˚ along the c-axis and 1.97 A ˚ with Zn–O distances of 1.99 A for the other three Zn–O distances, respectively. Because we do not expect geometrical relaxations for TM doped ZnO, we constructed a point charge field for the experimental structure of bulk ZnO in Wurtzite structure [42–46] with the nominal charges of +2 for Zn and 2 for O, in the same way as for our previous work on F centers in ZnO [41]. It is always important to make sure, that the electrons of the ab initio part cannot move into the point charge field. Therefore, all metal ions which are bound to O ions of the quantum cluster are explicitly included in the calculations. They were either described by large core pseudopotentials for Zn2+ from Stoll [47] without basis set. Such pseudopotentials are called total ion potentials (TIPs) [48,49]. Alternatively, we used an embedding by Mg2+ ions which were described by a single zeta basis set. The basis set was obtained from a calculation of Mg2+ in ZnO with the 8s5p basis set of Huzinaga [50]. For each Mg ion five

K. Fink / Chemical Physics 326 (2006) 297–307

basis functions, one for each occupied orbital, were included in the calculation. We call this approach ‘embedding by model atoms’. We used Mg2+ model atoms instead of Zn2+, in which 14 basis functions per ion were necessary, because this reduces the effort without changing the results. To prove the reliability of the cluster calculations, we compared our calculations on the mononuclear clusters to the experimentally observed absorption spectra and compared the results of both embedding procedures with each other. Most calculations were performed for the clusters, which were embedded by the Mg2+ model ions. If not explicitly mentioned the results are obtained with this embedding. We investigated clusters with one and two transition metal ions, respectively. The mononuclear cluster (cluster I) contained one metal atom, the four nearest O atoms, and 12 model atoms or TIPs. With the binuclear cluster (cluster II) the magnetic exchange coupling between two transition metal ions connected by an O bridge (see Fig. 1) was calculated. The bridging O was the central atom of the cluster. Furthermore, all four metal ions (two Zn and two TM ions) connected to the bridging O, the next shell of 12 O ions, and 25 model atoms or TIPs belonged to the quantum cluster. 3. Methods of calculation All calculations reported in the present paper were performed by means of the Bochum suite of quantum chemical ab initio programs [51–54]. The program package contains a restricted open-shell Hartree–Fock (ROHF) [51], a complete active space SCF (CASSCF) [52,53], and for the calculation of dynamic correlation effects a multi configuration coupled electron pair approximation (MCCEPA) program [54]. For the CASSCF wavefunctions, it is possible to calculate the spin orbit interaction and the Zeeman splitting [32] using finite perturbation theory. For the metal atoms, we used either small core pseudopotentials of the Stuttgart group [47] with the Stoll basis set or the all electron basis of Wachters [55] without the most diffuse s exponent, which is not necessary for ions with the charge +2.0. The Wachters basis was augmented by an additional semi diffuse d-function. The O atoms in the first shell were described by the 11s7p basis set of Huzinaga [50] contracted to 6s5p and augmented by a d-function with the exponent 0.8. A systematic description of the low-lying magnetic states of the binuclear complexes is obtained in the following manner: we first determined the orbitals for the high-spin state by either a ROHF calculation or a state average CASSCF calculation, in which all high-spin states with the correct number of d electrons were included. In this way it can be achieved that all d orbitals are partially occupied and all the ligand orbitals, i.e., the 2p orbitals at the O ions, are fully occupied. The ten partly occupied d orbitals of the binuclear complex are then localized by a Foster Boys localization [56], which yields localized orthogonal magnetic orbitals [57].

299

In the next step, the energies of the different spin states are obtained by a full CI in the space of the magnetic orbitals. Such a full CI in the active space of the CASSCF calculation is called CASCI. The main contributions to the wave function can be divided into covalent configurations, which contain the original number of d electrons at each center (dndn configurations), and ionic or charge transfer configurations, in which one electron is transferred from one of the TM centers to the other (dn+1dn1 configurations). A VCI calculation in the subspace of the covalent configurations leads always to a ferromagnetic coupling Jf, while the ionic configurations are responsible for the superexchange coupling JSE [58,59], which is normally antiferromagnetic, J ¼ J f þ J SE .

ð2Þ

Since the ionic configurations do not contribute to the high-spin states, the orbitals are optimized for the covalent configurations only. If these orbitals are also used for the ionic configurations, as it is done in the VCI calculations, their energies are much too high and, consequently, the superexchange contribution to J is too small. The situation is not much better if one performs a CASSCF calculation for the low-spin states, since the ionic configurations enter into them only with a rather small weight. It would be preferable to use ‘relaxed orbitals’ in the ionic configurations, i.e., orbitals which are adjusted to the charge transfer configuration TM3+TM+, however, that is not possible at the VCI level. A possibility to include ‘relaxed orbitals’ for the ionic configurations was used by Broer et al. [60], who performed a non-orthogonal CI calculations, with different sets of orbitals for the neutral and the charge transfer states. The error connected with the use of non-relaxed orbitals can be corrected by two approaches: inclusion of dynamic correlation by, e.g., the MCCEPA method [54], CASPT2 [61], or DDCI [62,63]. Several authors observed an increase of the superexchange coupling by inclusion of dynamic correlation effects. Examples are the calculations on NiO by de Graaf et al. [64] or on KNiF3 by Illas et al. [65] and in our own work, e.g., [30,31]. De Loth et al. [66] and recently Calzado et al. [67,68] analyzed in detail the different mechanisms, which contribute to the magnetic coupling. Calzado et al. [67,68] showed that U is dramatically reduced in the DDCI calculations and the relative weight of the ionic configurations increased by a factor of 2–5. Such a decrease of U was although observed at the DFT level by Martin and Illas [34]. A second possibility is a modified valence CI approach (MVCI). The orbital relaxation energy R of the ionic configurations can be estimated using the ionization potential (IP) and the electron affinity (EA) calculated for the mononuclear clusters: by calculating IP and EA for a mononuclear TM2+ center, first at the VCI level using the orbitals optimized for the TM2+ center and then at the CASSCF level both for a TM3+ and a TM+ center, we can extrapolate the relaxation energy of the ionic configurations as

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Fig. 1. Cluster I and cluster II. Black, small: Zn; gray: O; black, big: TM for cluster I. In cluster II the big black balls represent TM centers and the medium size balls Zn centers for coupling path I. For coupling path II, its vice versa.

R ¼ EðTM3þ ÞVCI EðTM3þ ÞCASSCF þEðTMþ ÞVCI EðTMþ ÞCASSCF .

ð3Þ The diagonal elements of the ionic configurations in the VCI for the binuclear cluster are shifted by the relaxation

energy R to lower energies. Technically, this is done in the following way: the magnetic orbitals are divided into two subspaces, which each contain the orbitals of one center. For these subspaces the occupations can be defined by occupation schemes. The first scheme defines the reference

K. Fink / Chemical Physics 326 (2006) 297–307

occupation, in our case the neutral configuration. The energies of all configurations which belong to other occupation schemes (ionic configurations) are shifted by the relaxation energy. The same shift as obtained for the singly ionized states was used for doubly or higher ionic configurations. The error occurring from this global shift is negligible, because these higher states contribute only with very small weights to the wave function. The results do not even change, when the doubly or triply ionic states are completely removed from the CI space. Because of the smaller energy denominator the weight of the ionic configurations in the CI wave function and the superexchange coupling are increased. The calculated J values have the quality of those obtained with inclusion of dynamic correlation at the cost of a CASSCF calculation [69]. A similar approach was used previously by Goodgame and Goddard for the calculation of the excites states of Cr2 and Mn2 [70]. In our calculations the energy difference U between covalent and ionic configurations is typically reduced by about 50% so JSE is doubled. All data included in the method is obtained from ab initio calculations. This method bears the further advantage that one can include the spin orbit coupling easily in the MVCI calculation. Spin orbit coupling (SOC) was included in the same manner as in our former work on octahedrally surrounded Co2+ centers [32,33]. This means that we scale the one electron SOC operator XX Za ~ ð4Þ H SOC ¼ s l~ 3 i i a i ðRa  r i Þ with an empirical scaling factor n in order to simulate the effects of the two electron SOOC (spin other orbit coupling) operator. Za is the charge of the nucleus a and li and si are the orbital angular momentum and the spin operators for the ith electron, respectively. We have always only one type of TM atoms in our complexes. Therefore, the scaling of the whole SOC operator is equivalent to the scaled nucleus SOC method. A discussion on different approximations to include spin orbit coupling was lately given by Neese [71]. While the scaled nucleus SOC approximation fails for heavy elements [72], it works quite well for 3d elements. We obtained the scaling factors by fitting the spin orbit splitting of the free TM2+ ions to the experimental values. For Co2+ we took n = 0.61 from our previous work, for Ni we obtained 0.60. The fitting for Mn was not possible because the ground state does not show spin orbit splitting and the higher states of the d5 configuration show such a complicated splitting that it cannot be fitted by one parameter. We performed calculations with the Ni value. This value is a bit to large if one fits n to the 4P state on Mn, where we obtained n = 0.48. But, as the ground state of Mn2+ is not influenced by spin orbit splitting, the choice of n is not really important. For the mononuclear clusters, we also calculated the g values by using finite perturbation theory and the microscopic Zeeman operator

HZ ¼

X

bð~ li þ g e ~ B; si Þ~

301

ð5Þ

i

where b is the Bohr’s magneton, ge the g factor of a free electron and ~ B is the strength of the magnetic field. In principle g is a tensor. In compounds without zero field splitting the components along the principle axes can be obtained from the energy change of an electronic state in a small magnetic field by gu ¼ 

DE ; bBu DM

ð6Þ

u denotes the component, DE is the energy difference with and without magnetic field. The situation is more complicated for the 4A2 ground state of Co2+ in a trigonal distorted surrounding. The splitting of the 4A2 state is given by Kahn for the equivalent situation of a Cr3+ ion in distorted octahedral surrounding [73]. The 4A2 state splits into two Kramers doublets E1/2 and E3/2, with an energy difference of 2D. The distortion is along the z-axis. For a magnetic field in z-direction the energies of the four components of the ground state are given by: E12 ¼ gk bBz =2;

ð7Þ

E32 ¼ 2D  3gk bBz =2.

ð8Þ

The directions x and y are degenerate and g? can be obtained by applying a magnetic field, e.g., in x-direction. For |D|  g?bBx the energies are given by E1;2 ¼ g? bBx  3g2? b2 B2x =8D; E3;4 ¼ 2D þ

3g2? b2 B2x =8D.

ð9Þ ð10Þ

4. Isolated Mn2+, Co2+, and Ni2+ atoms in ZnO Although we are mainly interested in the binuclear complexes, some information of the mononuclear complexes is needed for the further discussion. As mentioned above, we need the orbital relaxation energies for the IP and EA. Furthermore, it is important whether and in which way the electronic states of the TM2+ ions are influenced by spin orbit coupling. In ZnO each Zn2+ ion is tetrahedrally surrounded by O ions, with a slight distortion to C3v symmetry along the c-axis of the crystal. If a Zn2+ ion is substituted by a transition metal ion the electronic structure of the partly filled d shell can be described by ligand field theory for distorted tetrahedral complexes. In Td symmetry, the e orbitals are lower in energy than the t2 orbitals. The latter are splitted by the trigonal distortion into an e and an a1 component. The occupation of the d orbitals in the ground states of Mn2+, Co2+, and Ni2+ are summarized in Table 1. The ground states of Mn2+ and Co2+ are spatially nondegenerate, while Ni2+ has a 3T1 ground state which is split by the trigonal distortion into a 3E and a 3A2 state. The results of the calculations on the lowest state of Mn2+ are summarized in Table 2. The 6A1 ground state is well isolated from all other states. The spin orbit splitting

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spin orbit splitting of the electronic states of Co2+ is given in Table 4. We compared our results to the data of Weakliem [77] and Koidl [74]. For the absorption spectra of Co2+ in ZnO Weakliem observed two broad bands, the first centered at 3500 cm1 and the second at about 6000 cm1, which is assigned to the transitions from the 4A2 ground state of Co2+ to the 4T2 and 4T1 components of the 4F state of the free ion. The spin orbit coupling constant was fitted to k = 210 cm1. The total splitting between the spin orbit components of the 4T2 state is 2|k| = 420 cm1. The next states are located at about 15,000 cm1 and arise from a 2 G state of the free ion, while the 4T1 state, which originates from 4P is centered by 16,500 cm1. Weakliem neglected the trigonal distortion in his discussion. The results of Koidl [74], who considered the trigonal distortion, are included in Table 4. The 4A2 ground state is split by SOC into two Kramers doublets, separated by 5.4 cm1. The excitations from the ground state into the components of 4T2 are in the range between 3600 and 4100 cm1, i.e., in the same range as the results of Weakliem [77]. Additionally, several lines of the experimental spectrum were assigned to phonon side bands. The excitation from the ground state to the 4T1 state shows lines in the range of 6000–8000 cm1. Though one cannot expect to obtain very high accuracy for the excitation energies at the CASSCF level, Table 4 shows that the zero field splitting of the ground state and the energies of the spin orbit components of the 4T2 and

Table 1 Electronic states of different TM2+ centers in ZnO TM

Mn2+

Co2+

Ni2+

d Electrons e Occupation t2 Occupation Ground state in Td

5 2 3 6 A1

7 4 3 4 A2

8 4 4 3 T1

of this state is in our calculations about 0.02 cm1 and g = 2.002. This is identical to the pure spin state given the accuracy of our calculations. For the determination of the relaxation energy also the results for Mn3+ and Mn+ are included in Table 2. The ionization potential (IP) amounts to 12.00 eV and the electron affinity (EA) to 14.06 eV at the VCI level which means, that the energies of Mn3+ and Mn+ were calculated using the orbitals which were optimized for Mn2+. In individual CASSCF calculations for Mn3+ and Mn+ the IP was reduced by 4.49 eV and the EA by 5.51 eV, respectively. Therefore, we have to consider a relaxation energy of RMn = 10.00 eV for the ionic configurations of the binuclear cluster. The free Co2+ ion possesses two low-lying quartet states, 4 F and 4P, which are split by ligand field and spin orbit coupling. Table 3 contains our results for the electronic states of a single Co2+ ion in ZnO without inclusion of spin orbit coupling and also the results for Co3+ and Co+, from which we get the relaxation energy RCo = 11.44 eV. The Table 2 Electronic states Mn in ZnO Mn2+ State

Mn3+ 1

Energy (cm )

State

CASSCF 6

A1

a

0

IPa in eV 5 E 5 A1 5 E DErelax

Mn+ 1

Energy (cm )

State

VCI

CASSCF

12.00 0 94 5404

7.46 0 69 6397 4.49 eV

EAa in eV 5 E 5 E 5 A1 DErelax

Energy (cm1) VCI

CASSCF

14.06 0 2760 2975

8.61 0 1904 2320 5.51 eV

Energy of the lowest state of the ion relative to the Mn2+ ground state energy of 3869.206214 H.

Table 3 Electronic states of Co in ZnO Co2+ State

Co3+ Energy (cm1)

State

CASSCF A2 4 A1 4 E 4 A2 4 E 4 E 4 A2

0 3351 3358 5629 6026 21,950 22,272

IP in eV 5 E 5 A1 5 E

DErelax a

Energy (cm1) VCI

a

4

Co+

12.07 0 4382 4440

State CASSCF 6.58 0 4437 4467

5.48 eV

a

EA in eV 3 E 3 A2 3 E 3 A1 3 A2 3 A2 3 E DErelax

Energy of the lowest state of the ion compared to the Co2+ ground state energy of 4100.752625 H.

Energy (cm1) VCI

CASSCF

12.22 0 355 1876 1946 3989 19,345 19,498

6.32 0 411 1989 2118 4195 17,694 17,764 5.96 eV

K. Fink / Chemical Physics 326 (2006) 297–307 Table 4 Spin orbit coupling of Co2+ ions in ZnO CASSCF

SO–CI

State

State

Energy (cm1)

Exp. Energy (cm1)

Energy (cm1)a

4

A2

0

E12 E32

4

A1

3351

4

E

3358

E12 E32 E12 E12 E12 E12

3157 3173 3327 3598 3634 3708

0 15 170 440 476 551

4

A2

5629

4

E

6026

E12 E32 E12 E12 E32 E12

5373 5768 5959 6440 6621 6716

0 395 586 1067 1248 1342

a b

0.0 5.8

0.0 5.8

Energy (cm1)b

Energy (cm1)a,b

0

0 5.4 0 19 99 309 340 445

3611–4057

0 181 657 781 1149 1492

5912–7504

Relative to the lowest state of the electronic configuration. Experimental data from Koidl [74].

the first 4T1 state are correctly reproduced in our calculations. Also the calculated values of the g tensor of the ground state, gk = 2.33 and g? = 2.39, are in good agreement with the experimental data of gk = 2.24 and g? = 2.28 [78]. While gk could be easily obtained with field strength of 0.01 and 0.005 a.u., we had to use fields of 0.0005 and 0.001 a.u. to obtain g?, because of the condition |D|  g?bBx (B = 1 a.u. = 1.715 · 103 T). We calculated g? from the linear contribution 2g?bBx, which is the difference of the energies E3 and E4 in Eq. (10). Only the positions of the excited states which arise from the 4P state of the free ion are on error by about 5000 cm1, but this is of no importance for the present purpose, since for the description of the magnetic exchange coupling it is only necessary to start from a reliable electronic wave function for the Co2+ ground state. The situation is more complicated for Ni2+ ions in ZnO. The results without inclusion of spin orbit coupling are summarized in Table 5. We obtained the relaxation energy

303

RNi = 12.05 eV for the charge transfer configurations from these data. The ground state of Ni2+ in tetrahedral surrounding has 3T1 symmetry, but it is split by the trigonal distortion into a 3E and a 3A2 state. This picture changes dramatically, as soon as spin orbit coupling is taken into account [75–77]. The results of our calculations and the experimental data, which were obtained by measurements of the magnetic susceptibility [75] and analysis of the fine structure of the absorption spectra [76], are listed in Table 6. The splitting of the 3T1 state by spin orbit coupling is comparable to the splitting of a 3P state of an atom into J = 0, 1, 2 fine structure levels. The ground state is an A1 state, which has, comparable to a J = 0 state, no first order interaction with the magnetic field. The next states correspond to J = 1, but are further split by the trigonal distortion into an A2 and an E state. The highest states listed in Table 6 correspond to J = 2. Except for the position of the A2 state, our results are in good agreement with the data of Anderson [76]. The first state exhibiting a first order Zeeman splitting is the lowest E state, which should only be populated at higher temperatures. This fact is discussed in the experimental work of Gamelin and coworkers [79,15]. These authors found a dramatic decrease of the paramagnetism of the Ni2+ centers for temperatures below 100 K. We have considered whether the without spin orbit coupling spatially degenerate 3E ground state of Ni2+ is split by a Jahn–Teller distortion in such a way that the true ground state is no longer spatially degenerate. We checked

Table 6 Spin orbit splitting of the 4T1 state of Ni2+ in ZnO (CASSCF) CASSCF State

SO–CI 1

Energy (cm )

3

0

3

301

E A2

Exp. 1

State

Energy (cm )

Ref. [75]

Ref. [76]

A1 A2 E E E A1

0 250 403 967 1274 1374

0 160 260

0 140 372 863 1295 1421

Table 5 Electronic states of Ni in ZnO Ni2+ State

Ni3+ Energy (cm1)

Ni+ Energy (cm1)

State

CASSCF 3

E 3 A2 3 E 3 A1 3 A2 3 A2 3 E

0 301 2560 2617 5572 20,843 21,100

IPa in eV 2 A1 2 E 2 E

VCI

CASSCF

13.22 0 86 1852

7.36 0 168 2353

5.83 eV

DErelax a

State

2+

Energy of the lowest state of the ion compared to the Ni

EAa in eV 4 A2 4 A1 4 E 4 A2 4 E 4 E 4 A2 DErelax

ground state energy of 4226.185620 H.

Energy (cm1) VCI

CASSCF

11.65 0 4323 4350 7332 7745 24,160 24,620

5.39 0 4512 4532 7675 8235 24,350 24,848 6.22 eV

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this possibility but the influence of spin orbit coupling was always the dominant effect. 5. Exchange coupling of two TM centers For the calculation on the exchange coupling between two TM centers, the binuclear cluster II has been chosen. Generally, there are two slightly different coupling paths. Path I is the coupling of two TM centers in the same layer of the Wurtzite lattice and path II the coupling between TM ions of different layers. The simplest case is that of the coupling between two Mn2+ ions in their 6A1 ground states. We started from a ROHF calculation for the high-spin, 2S + 1 = 5, state of the cluster with two Mn2+ centers. All 3d orbitals at both Mn centers are singly occupied, and therefore, they are all described with the same accuracy. The d-orbitals were then localized and we performed CASCI calculations in the space of the 3d orbitals to obtain the energies of the lower spin states. In these CI calculations the energies of the ionic configurations were lowered by the relaxation energy RMn, which had been obtained in the calculations for the mononuclear clusters. The results for the two different coupling paths are summarized in Table 7. In both cases, the Mn2+ ions are only weakly coupled. The coupling is antiferromagnetic, with J being 14.1 and 10.8 cm1 for the two different coupling paths. The splittings between the different spin states exhibit a nearly perfect Lande´ pattern. It is not necessary to consider spin orbit coupling because the Mn2+ ions posses no orbital momentum. We checked our results with a cluster embedded by Zn large core pseudopotentials and obtained a coupling constant of 15.0 cm1 for coupling path I. This gives some confidence to the reliability of our embedding. The result is in good agreement with the experimental value of J = 10.5 cm1 of Fukumura et al. [6]. The coupling between two Co2+ ions is quite similar to the coupling between the Mn2+ centers (see Table 8). Without spin orbit coupling included, the exchange coupling is weakly antiferromagnetic with J = 11 and 4.7 cm1 for path I and path II, respectively. For the embedding in TIPs we obtained with J = 10 cm1 and J = 9 cm1 comparable results. As shown in the section on the mononuclear clusters, the 4A2 ground state of one Co2+ ion exhibits only a small spin orbit splitting of 5.8 cm1 into

Table 7 Exchange coupling between two Mn2+ centers in ZnO State

S=0 S=1 S=2 S=3 S=4 S=5

Path I

Path II

Energy (H)

J (cm1)

Energy (H)

J (cm1)

11876.467147 11876.467019 11876.466762 11876.466377 11876.465863 11876.465216

14.1 14.1 14.1 14.1 14.2 14.2

11876.467059 11876.466961 11876.466765 11876.466471 11876.466079 11876.465588

10.8 10.8 10.8 10.8 10.8 10.8

two Kramers doublets. Table 8 shows that this splitting does not seriously change the order of the electronic states of the binuclear complex. It leads only to small splittings of the triplet, quintet and septet states. As for Mn, we obtained a Lande´ splitting between the different spin states. We conclude from our results that also in case of Co2+, it is not necessary to consider spin orbit coupling. This is in contrast to the results of de Graaf and Sousa on the magnetic exchange coupling in Co2 Cl2 [80], where the two 6 Kramers doublets at each Co center are split by 30 cm1 and the electronic states of the dinuclear complex cannot be explained by a Lande´ splitting. The results of our calculations for the interaction of two Ni2+ centers are summarized in Tables 9 and 10. Without inclusion of spin orbit coupling we obtained four quintet, triplet and singlet states in a range of about 170 cm1. The ground state is a quintet, but the energy levels do not at all show a Lande´-type behavior. It is not possible to extrapolate J values from the electronic energy levels. The low-lying states are generated by the coupling between the lowest 3E states at each center. The next states follow at about 340 cm1, they are produced by the coupling of the 3 E state at one Ni with the 3A2 state at the other Ni center. Inclusion of spin orbit coupling totally changes the picture. From the investigations on the mononuclear clusters, we

Table 8 Exchange coupling between two Co2+ centers in ZnO Spin

Without spin orbit coupling Energy (H)

DE (cm1)

Coupling path I S = 0 12339.559876

0.0

S=1

12339.559777

21.7

S=2

12339.559580

S=3

With spin orbit coupling J (cm1)

Degeneracy

DE (cm1)

1

0

10.9

2 1

19.3 27.1

65.0

10.8

2 1 2

63.9 64.6 65.3

12339.559286

129.5

10.8

1 1 1 2 2

122.3 123.3 124.1 127.6 134.5

Coupling path II S = 0 12339.559445

0.0

1

0

S=1

12339.559402

9.5

4.7

2 1

8.0 14.6

S=2

12339.559316

28.4

4.7

2 1 2

28.4 28.8 30.4

S=3

12339.559189

56.3

4.7

2 1 1 1 2

51.7 55.1 56.6 57.5 62.1

K. Fink / Chemical Physics 326 (2006) 297–307

305

Table 9 Exchange coupling between two Ni2+ centers in ZnO Path I

Path II 1

DE (cm1)

Spin

Energy (H)

DE (cm )

Spin

Energy (H)

5 1 3 1 3 1 3 5 5 3 5 1

12590.425865 12590.425734 12590.425693 12590.425519 12590.425443 12590.425425 12590.425373 12590.425316 12590.425149 12590.425118 12590.425091 12590.425085

0.0 28.7 37.7 75.9 92.6 96.5 107.9 120.4 157.1 163.9 169.8 171.1

5 3 1 1 5 3 1 3 5 5 1 3

12590.424978 12590.424966 12590.424963 12590.424905 12590.424893 12590.424888 12590.424732 12590.424722 12590.424709 12590.424709 12590.424687 12590.424619

0.0 2.6 3.3 16.0 18.7 19.8 54.0 56.2 59.0 59.0 63.9 78.8

1 1 5 3 3 5 1 1 3 3 5 5

12590.424315 12590.424306 12590.424272 12590.424253 12590.424226 12590.424217 12590.424031 12590.423923 12590.423920 12590.423919 12590.423704 12590.423596

340.1 342.1 349.6 353.7 359.7 361.6 402.5 426.2 426.9 427.0 474.2 497.9

5 5 1 3 3 1 5 1 3 1 3 5

12590.424516 12590.423464 12590.423383 12590.423373 12590.423362 12590.423347 12590.423320 12590.423188 12590.423073 12590.422887 12590.422843 12590.422821

101.4 332.3 350.1 352.3 354.7 358.0 363.9 392.9 418.1 458.9 468.6 473.4

1 3 5

12590.422816 12590.422703 12590.422474

669.1 693.9 744.2

1 3 5

12590.421620 12590.421577 12590.421461

737.0 746.4 771.9

know that the ground state of each Ni ion is a non-magnetic A1 state followed by an A2 state at 250 cm1 and an E state at 400 cm1. The ground state of the binuclear complex arises from the coupling of the two local A1 states at the Ni centers and is followed by the plus and minus combina-

Table 10 Coupling of two Ni centers in ZnO with inclusion of spin orbit splitting States

Path I Energy (H)

Path II 1

DE (cm )

Energy (H)

DE (cm1)

A1 · A1

12590.433478

0.0

12590.432741

0.0

A1 · A2

12590.432415 12590.432247

233.4 270.1

12590.431763 12590.431636

214.5 242.5

A1 · E

12590.431800 12590.431765 12590.431507 12590.431504

368.4 376.0 432.6 433.2

12590.431021 12590.430899 12590.430886 12590.430725

377.5 404.1 407.1 442.3

A2 · A2

12590.431157

509.4

12590.430617

466.1

A2 · E

12590.430632 12590.430601 12590.430389 12590.430383

624.6 631.5 678.1 679.2

12590.430037 12590.429817 12590.429799 12590.429688

593.3 641.6 645.7 670.0

E·E

12590.430057 12590.429778 12590.429714 12590.429547

750.9 812.0 826.2 862.9

12590.429149 12590.429117 12590.428960 12590.428856

788.2 795.0 829.7 852.5

Fig. 2. Electronic states of the binuclear Ni complex, coupling path I, the lowest states do not show a first-order interaction with the magnetic field.

tions of the local A1 and A2 states at 230 and 270 cm1. The next states follow at about 400 cm1. These states are the first, which show a first order Zeeman splitting (see Fig. 2). We can draw the following conclusions: It does not make any sense to analyze the magnetic properties of Ni doped ZnO without accounting for spin orbit coupling. After inclusion of spin orbit coupling the Ni2+ ions are diamagnetic and cannot show a magnetic exchange coupling at low temperatures. This was mentioned already by Anderson [76] in 1967. The paramagnetism at higher temperatures can be caused by an occupation of the lowest E state at a local Ni center. In the binuclear complex, the lowest E state which shows a first-order interaction with the magnetic field is generated by a linear combination of the A1 ground state at one Ni2+ center with the E state at the other. Based on our calculations this costs 432 cm1, which is in the same order as the energy of the E state in the mononuclear cluster. Even if we take into account the lower experimental values of Brumage and Lin [75] those states are still about 260 cm1 above the ground state. Even this is too high to take place at room temperature. Our results are in good agreement with the experimental data and the interpretation of Radovanovic and Gamelin [15]. 6. Conclusions In this paper, we investigated the magnetic exchange interaction of TM doped ZnO (TM = Mn2+, Co2+, and Ni2+) by embedded cluster calculations and could prove that embedded cluster methods are appropriate for a detailed analysis of the magnetic properties of TM doped ZnO. We started by investigations on ZnO clusters, which contained only one transition metal ion. Our cluster model correctly describes the electronic structures of the ground states and the d–d excited states of the transition metal centers. For Ni2+ in ZnO it was necessary to include spin orbit coupling to obtain a reasonable description of the lowlying electronic states. On the other hand, spin orbit coupling is negligible for Mn2+ and causes only a small zero field splitting for Co2+.

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For the binuclear complexes, we limited our calculations on the interaction between nearest neighbors. For Mn as well as for Co we found an antiferromagnetic coupling with J values between 8 and 15 cm1 in agreement with experiment and part of the supercell DFT calculations. The Heisenberg Hamiltonian is valid and we obtain Lande´ splittings between the different spin states. For Ni it proved to be necessary to consider spin orbit coupling, because it changed the properties of the electronic states completely. Already without spin orbit coupling the electronic states do not show a Heisenberg behavior. Therefore, this system should not at all be described by calculations, which are based on a simple Heisenberg Hamiltonian. With inclusion of spin orbit coupling the Ni centers are diamagnetic at low temperatures and cannot show any magnetic interaction. At higher temperatures an E state can be occupied, which is responsible for the observed paramagnetism [15]. In none of the three cases the magnetic exchange coupling of the TM2+ centers can be the origin of room temperature ferromagnetism. Therefore, we plan an embedded cluster study on the double exchange between TM+ and TM2+, and TM3+ and TM2+, respectively. Acknowledgement The work was supported by the Deutsche Forschungsgemeinschaft through the Sonderforschungsbereich 558 ‘‘Metall-Substrat-Wechselwirkung in der heterogenen Katalyse’’ and the COST D26 working group ‘‘Theoretical understanding and prediction of magnetic properties in molecules and solids’’. References [1] J.K. Furdyna, J. Appl. Phys. 64 (1988) R29. [2] T. Dietl, H. Ohno, F. Matsukura, J. Cibert, D. Ferrand, Science 287 (2000) 1019. [3] C. Zener, Phys. Rev. 81 (1951) 440. [4] S.J. Pearton, W.H. Heo, M. Ivill, D.P. Norton, T. Steiner, Semicond. Sci. Technol. 19 (2004) R59. [5] W. Prellier, A. Fouchet, B. Mercey, J. Phys.: Condens. Matter 15 (2003) R1583. [6] T. Fukumura, Z. Jin, M. Kawasaki, T. Shono, T. Hasegawa, S. Koshihara, H. Koinuma, Appl. Phys. Lett. 78 (2001) 958. [7] C.N.R. Rao, F.L. Deepak, J. Mater. Chem. 15 (2005) 573. [8] S.W. Jung, S.-J. An, G.-C. Yi, C.U. Jung, S. Lee, S. Cho, Appl. Phys. Lett. 80 (2002) 4561. [9] P. Sharma, A. Gupta, F.J. Owens, A. Inoue, K.V. Rao, J. Mag. Mag. Mat. 282 (2004) 115. [10] K.R. Kittilstved, N.S. Norberg, D.R. Gamelin, Phys. Rev. Lett. 94 (2005) 147209. [11] K. Ueda, H. Tabata, T. Kawai, Appl. Phys. Lett. 79 (2001) 988. [12] H.-J. Lee, S.-Y. Jeong, C.R. Cho, C.H. Park, Appl. Phys. Lett. 81 (2002) 4020. [13] D.P. Norton, M.E. Overberg, S.J. Pearton, K. Pruessner, J.D. Budai, L.A. Boatner, M.F. Chisholm, J.S. Lee, Z.G. Khim, Y.D. Park, R.G. Wilson, Appl. Phys. Lett. 83 (2003) 5488. [14] A.S. Risbud, N.A. Spaldin, Z.Q. Chen, S. Stemmer, R. Seshadri, Phys. Rev. B 68 (2003) 205202. [15] P.V. Radovanovic, D.R. Gamelin, Phys. Rev. Lett. 91 (2003) 157202.

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