Synthesis and magnetic behaviour of Mn:ZnO nanocrystalline powders

Spectrochimica Acta Part A 75 (2010) 1218–1222

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Synthesis and magnetic behaviour of Mn:ZnO nanocrystalline powders R.Vidya Sagar ∗ , S. Buddhudu Department of Physics, Sri Venkateswara University, Tirupati 517502, Andhra Pradesh, India

a r t i c l e

i n f o

Article history: Received 15 June 2009 Received in revised form 22 November 2009 Accepted 2 December 2009 PACS: 75.50.Pp 61.72.uj 76.50.+g 61.05.Cp

a b s t r a c t This paper reports on the magnetic properties of Mn:ZnO nanoparticles. XRD profiles have shown that the undertaken materials are in wurtzite structures. The crystallite size of the sample has been examined using TEM for one sample. In order to verify the lattice site occupancy and also valence state of the manganese ion, EPR spectral measurements have also been carried out for these samples. The magnetic properties of the samples have been investigated on a Vibrating Sample Magnetometer (VSM). © 2009 Elsevier B.V. All rights reserved.

Keywords: ZnO: Mn Nanoparticles Ferromagnetism XRD

1. Introduction An extensive research work has been going on in a rapid manner on semiconductor materials which posses ferromagnetism above the room temperature. To meet this requirement, there should be either a transition metal or a rare earth element to be present in the semiconductors. Earlier, studies have been focused on III–V compounds that are based diluted magnetic semiconductors (DMSs) which have attracted more interest due to their applications in optoelectronic and high-speed IC devices development. The equilibrium solubility of magnetic impurities in III–V semiconductors is very low, and the existing magnetic impurities tend to segregate and form secondary phases. Thus an ideal density of magnetic impurities could not be obtained by using a conventional equilibrium crystal growth method. The first successful non-equilibrium DMS epilayer growth has been demonstrated by Munekata et al. [1] and they have used a low-temperature MBE method to achieve an epitaxial growth of InMnAs on GaAs substrate and thus they have successfully suppressed the surface segregation and also second phase formation. This, in turn, led to the discovery of ferromagnetism in p-type InMnAs at low temperatures. In 1996, an epitaxial growth of GaAs-

∗ Corresponding author. Tel.: +91 0877 2261611; fax: +91 877 2249273. E-mail address: [email protected] (R.Vidya Sagar). 1386-1425/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.saa.2009.12.014

based DMS specifically GaMnAs films has been demonstrated and ferromagnetism has also been reported below room temperature [2]. An important finding has been the direction of magnetization here. It turns out with an easy axis of magnetization in GaMnAs grown on GaAs (0 0 1) plane that has been oriented within the plane due to the compressive strain since the lattice constant of GaMnAs is slightly larger than that of GaAs. One can easily enhance or reduce the effect of strain by simply introducing a proper buffer layer. Most of the earlier attentions have mostly been focused on ferromagnetic semiconductors of (In,Mn)As and (Ga,Mn)As systems [3–7]. A great deal of research on (Ga,Mn)As and (In,Mn)As materials has been carried out with some interesting results, such as long spin lifetime and the ability to achieve spin transfer through a heterointerface, either in semiconductor–semiconductor or metal–semiconductor synthesis [7–10]. Ever since the prediction of room temperature ferromagnetism has been realised from the wide bandgap semiconductors, several attempts have so far been made to produce ferromagnetism in Zn1−x Mnx O and Zn1−x Cox O materials using pulsed laser deposition technique [11–13], ion implantation [14,15], molecular beam epitaxy [16], and solid-state reaction [17,18]. There are several other reports on ferromagnetic behaviour in the Zn1−x TMx O systems (where TM = Mn, Co, and Fe) [19,20]. Ferromagnetic behaviour is largely a function of the processing conditions; often, highly nonequilibrium methods that are still needed to produce ferromagnetic material [21]. There are also several reports on Zn1−x Mnx O and Zn1−x Cox O without ferromagnetic behaviour and recent results

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have shown evidence that ferromagnetism in Zn1−x Mnx O has not been inherent property of the system but it could be due to the formation of secondary phases or clusters [22]. For ZnO, it is possible to develop good quality powder samples using an oxalate precursor decomposition method. The predicted solubility limit of transition metal ions (TM) in ZnO depends on its concentration. However, we have made an attempt to verify the suitability of oxalate precursor decomposition method to produce good quality ferromagnetic Zn1−x TMx O material. 2. Experimental studies 2.1. Reagents and chemicals used All the reagents and the chemicals used were of analytical grade (99.9% purity) those were purchased from M/s Merck Chemical Company, Germany in the forms of acetates of Zn and Mn that were dissolved in anhydrous ethanol at the room temperature. Ethanol was used as a solvent because of its dehydrating property and an oxalic acid in ethanol was used as a precipitating agent. 2.2. Synthesis of ZnO:Mn2+ nanoparticles Zn1−x Mnx O nanoparticles were prepared from a starting solution of zinc acetate dihydrate [Zn(ac)2 ·2H2 O] and manganese acetate tetrahydrate [Mn(ac)2 ·4H2 O] acetates in ethanol. Nanocrystalline bulk Zn1−x Mnx O nanoparticles were prepared using single source oxalate [Zn1−x Mnx (C2 O4 )·2H2 O] precursor decomposition method with x = 0, 0.01, 0.02, 0.04 and 0.08 mol% respectively. Due to the inherent metastability of the Zn1−x Mnx O system, crystalline oxalates are suitable precursors, upon intimate mixing of Zn2+ and Mn2+ ions on lattice sites prior to decomposition, with the removal of carbon as CO and CO2 during the decomposition process, leaving a phase pure oxide product. Oxalates were first prepared by mixing 0.4 M solutions of zinc acetate dihydrate [Zn(ac)2 ·2H2 O] and manganese acetate tetrahydrate [Mn(ac)2 ·4H2 O] mixed with 0.4 M solution of oxalic acid. The de-hydrating capability of ethanol plays an important role in the removal of coordinated water from the precursor compounds that promote the formation of oxalate structure at the room temperature. The precipitates thus formed were separated from the solution by filtration and those were collected and rinsed extensively in a double distilled water then those were dried in air at 333 K. These oxalates were powdered and decomposition of these oxalates into oxides was obtained at 673 K for 30 min in air. 2.3. Physical property measurements The present structural studies were performed using Xray diffractometer (Model: Siemens D-5000) using the Cu K␣ ( = 1.5405 Å) radiation. All measurements were carried out at 40 kV with 30 mA in the range of 20◦ ≤ 2 ≥ 80◦ . The scanning step and sampling time were 0.02 and 1 s, respectively. The average crystallite size was estimated using transmission electron microscope (Model: JEOL JFC-1000E), sample was gold sputtered at 15 kV for 200 s in vacuum (10−3 Torr) for performing TEM measurement. ESR (Model: Varian E-112) spectra were recorded to understand the nature of point defects, chemical state. The magnetization measurements were made on a Vibrating Sample Magnetometer (Model No. EV9 manufactured by ADE Magnetics, Wilson Way Westwood, MA 02090, USA), with the magnetic field both in parallel and perpendicular directions to the plane of the samples. The high precision stepped field control used in conjunction with digital signal averaging leading to an increasing accuracy of the measured graphs and measurement parameters, while at the same time facilitating measurements on soft magnetic samples with a

Fig. 1. XRD profiles of reference ZnO and Zn1−x Mnx O nanoparticles.

field resolution better than 0.01 Oe. Since the field was stepped up rather than swept, there is a higher accuracy in the field setting and a more precise determination becomes possible coercivity and many other magnetic parameters. 3. Results and discussion 3.1. X-ray diffraction of Zn1−x Mnx O nanoparticles Fig. 1 shows the powder X-ray diffraction profiles of Zn1−x Mnx O nanopowders with x = 0, 0.01, 0.02, 0.04 and 0.08 respectively. The X-ray diffraction peaks have been indexed and found to be hexagonal wurtzite structured. The XRD profiles demonstrate nine prominent peaks, indicating a non-preferential-orientation, and it is evident that those peaks correspond to the wurtzite structure. In the XRD profiles, a peak centered at 2 = 59.5◦ was not observed, which was attributed to the precursor compound. This precursor phase could be due to basic zinc acetate or hydroxide [23,24]. No trace of manganese, oxides or any binary zinc manganese phase has been observed in any of the samples up to 8% Mn adding. It could be confirmed exactly that Mn could be doped into ZnO matrix by substituting Zn with Mn. The evolution of unit cell parameters as function of Mn content has been given in Table 1. The lattice parameters for hexagonal ZnO and Mn-doped ZnO nanoparticles were estimated from the equation: 1 dh2 k l

=

3(h2

4 12 + 2 2 2 c + hk + k /a )

where a and c are the lattice parameters and h, k, and l are the Miller indices and dh k l is the interplaner spacing for the plane (h k l). This Table 1 The unit cell parameters (a, c, V and crystallite size D) of undoped and Mn doped ZnO nanoparticles. S. no.

Material

a (Å)

c (Å)

V (Å3 )

D (nm)

1 2 3 4 5

Reference ZnO Zn0.99 Mn0.01 O Zn0.98 Mn0.02 O Zn0.96 Mn0.04 O Zn0.92 Mn0.08 O

3.2414 3.2435 3.2456 3.2490 3.2578

5.2008 5.2040 5.2068 5.2118 5.2244

47.3196 47.4112 47.4973 47.6435 48.0183

16.59 15.34 14.16 12.34 9.23

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interplaner spacing can be calculated from: 2d = sin  = n where  is the wavelength of X-ray,  is diffraction angle, and n is the order diffraction (n = 1). However, the general trend shows that as the a and c parameters are increasing with an increase of Mn content and which follow the Vegard’s law [25–27] in the entire range studied. The tetrahedral ionic radius of Zn2+ is 0.74 Å, which is small compared to 0.83 Å for Mn2+ . Due to this fact, it could be noticed that lattice parameters a and c increase as function of Mn content change. If Mn2+ were in an octahedral environment, it could have been reflected as a dramatic change (decrease in lattice parameter a and c) in the lattice parameter as a function of x; octahedral Mn2+ has an effective ionic radius between 0.68 Å (low spin), 0.72 Å (high spin), against 0.74 Å for Zn2+ . Therefore, the Mn ions are understood to have occupied the Zn sites without changing its wurtzite structure. The volume of the unit cell for a hexagonal system has been calculated from the following equation: V = 0.866 × a2 × c The unit cell volume increases for single-phase compositions as lattice parameters a and c change. Incorporation of Mn ions into the ZnO lattice could easily be obtained from the lattice constants, which are from Vegard’s law. The intensity of the XRD peaks could decrease as a function of the Mn content change. Among the nine peaks, few are found to be overlapping because of the fact that FWHM broadening could occur. This trend reveals could show nanocrystalline nature of the precipitates. The average crystallite size D was estimated from the Debye–Scherrer’s equation [28]: Dh k l =

K ˇ cos 

In this equation,  represents the wavelength of the X-ray radiation Cu K␣ ( = 1.5405 Å), ˇ is the full width at half maximum of the diffraction peak (in rad) and  is the maximum of the peak (in rad) at maximum scatter angle (). The crystallite sizes are obtained from the Scherrer’s relation, only after applying appropriate background correction. The distribution of crystallite size of samples can be a possible reason resulting in broadening diffraction peaks. We use the (1 0 2) peak in the XRD patterns to calculate the average crystallite size. The obtained results of crystallite sizes are given in the following (Table 1) as a function of the Mn content. The TEM image shows that the crystallite size of Zn0.99 Mn0.01 O is ∼15 nm in Fig. 2 and is in good agreement with the XRD results.

Fig. 2. TEM image of ZnO nanoparticles.

tion, where the A0 (d4 ) center traps tightly an electron in the 3d shell forming the high spin, S = 5/2, 3d5 configuration, and this negatively charged Mn ion binds the hole in an effective mass state. In our studies, the divalent state (Mn2+ ) on the lattice site is observed, as evidenced in literature from Zn1−x Mnx O [31–33]. We have obtained g value in the Zeeman interaction term to be ∼2.0071; it is assigned to isolated Mn2+ ions substitutionally incorporated in the nanoparticals. For the studied Zn1−x Mnx O with concentration (x = 0.01, 0.02, 0.04 and 0.08), the integrated density increases linearly with the Mn content change. From the nanocrystalline ZnMnO samples, we could find Mn ions doped ZnO showing one broad signal without its normal six-line structure, regardless of x values. The broad signal without six-line structure is typically a demonstration of exchange-coupled Mn ions. This observation confirms that Mn2+ substitutes randomly the Zn2+ ions with no nearest–neighbour antiferromagnetic interactions. From Fig. 3, it is evident that as Mn concentration increases, i.e., decreasing Mn separation, the peak-to-peak width decreases due to exchange narrowing of the ESR signal. The typical exchange

3.2. Magnetic properties 3.2.1. ESR measurements of Zn1−x Mnx O nanopowders Fig. 3 show the ESR spectra of Zn1−x Mnx O (where x = 0.04 and 0.08) recorded at room temperature. The ESR spectra were performed at 9.3 GHz with a Microwave power 10,000 mW. The three possible electronic states of the Mn impurity substituting a divalent cation: A0 (d4 ) and A0 (d5 + h) for Mn2+ , and A− (d5 ) for Mn2+ [29,30,33]. A0 denotes the neutral center, A− is negatively charged center, and the notation in parenthesis is the electronic configuration of the d electrons. There are no reports on the observation of A0 (d4 ) neutral centers in ZnO. In the case of the A0 (d4 ) center, the hole resides in the 3d shell. In contrast, the anisotropy of some of ESR lines in the illuminated n-type ZnO can be explained in terms of A0 (d4 ) centers that undergo a Jan–Teller distortion, as was observed for Cr (3d4 ). However, a strong Hund’s rule intra-site exchange interaction may favor a state having 5d electrons and a loosely bound hole. This is the case with A0 (d5 + h) configura-

Fig. 3. ESR spectra of Zn1−x Mnx O nanoparticles.

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Table 2 Lande ‘g’ factor, line width (H) and number of spins (Ns ) of Zn1−x Mnx O powder samples. Concentration of Mn

g-Value

Line width, H (G)

No. of spins, Ns (×104 )

0.01 0.02 0.04 0.08

2.0091 2.0072 2.0112 2.0196

502.8876 500.1107 477.2364 473.1183

2.3824 2.7660 4.3707 7.0340

coupled Mn ions [32] are observed in other II–VI nanocrystals, with increasing Mn concentrations, i.e., decreasing Mn separation. The ‘g’ obtained in the present work ∼2.00 confirm the paramagnetic sate of Mn2+ ions at room temperature. The calculated ESR data is presented in Table 2 and from this table, it is observed that the linewidth with the Mn concentration change decrease which could be due to an exchange narrowing of the ESR signal. The substitution of Mn2+ sites further confirms the XRD data where an increase in unit cell volume is observed for Mn2+ ion substituted sample. The number of spins (Ns ) participating in the resonance can be calculated by using the formula: Ns = 0.285I(H)

2

where ‘I’ is the peak-to-peak height (arbitrary units) and H is the line width (in G). As mentioned above as the Mn concentration (x) is increased the resonance signal becomes broader and hence the number of spins are increased. From Table 2, it is evident that the relation between the concentration and the number of spins are found to be linear. As concentration increases, the number of spins could increase due to an increase in absorption. The number of spins, the values of g and the line width for all samples are calculated using Rubbins and Bleanny relation [34]. 3.3. Magnetization measurements for ZnO and ZnO:Mn Magnetic behaviour of Zn1−x Mnx O nanoparticles with a nominal concentration x = 0, 0.04 and 0.08 has been studied at room temperature (RT) in the range of the magnetic field 0–25 kOe using a Vibrating Sample Magnetometer. Fig. 4(a) shows the dependence of magnetization with an applied magnetic field for the reference ZnO nanoparticles. A typical diamagnetic behaviour has been observed in the reference ZnO. The diamagnetic behaviour of reference ZnO is due to the unpaired electrons of its d orbital, which is responsible for the absence of a permanent magnetic moment. Then, when electrons are paired together, their opposite spins could cause the magnetic fields to cancel with each other. Accordingly, when an applied magnetic field is acting then a distortion in balance takes place in their orbiting and thus create small magnetic dipoles within the atoms, which oppose the applied field. This action produces a negative magnetic effect. Fig. 4(b) shows the magnetization versus applied magnetic field curve for x = 0.04. The measurement reveals that the sample is in paramagnetic behaviour. Observed paramagnetic behaviour can be attributed to the actual incorporation of Mn ions in diamagnetic ZnO structure, although the precipitation of an antiferromagnetic and ferromagnetic phases could not be ruled out. However, these characteristics of Mn-doped ZnO samples have not been well known experimentally so far. The absence of ferromagnetic ordering in our results is found to be in agreement with the Zn1−x Mnx O sample with x = 0.36 was reported earlier in literature [35]. Among the available few reports, an apparent disagreement prevails with the magnetic properties of Mn doped ZnO that are expected to be strongly influenced by carrier type (p or n) and carrier density [36]. Sato and Yoshida [29] have predicted that the p-type (ZnMn)O

Fig. 4. (a) Magnetization vs applied magnetic field for reference ZnO nanoparticles. (b) Magnetization vs applied magnetic field for Zn0.96 Mn0.04 O nanoparticles. (c) Magnetization vs applied field for Zn0.92 Mn0.08 O nanoparticles.

could be ferromagnetic and n-type (ZnMn)O could be antiferromagnetic. The magnetic behaviour of Zn0.92 Mn0.08 O sample has considerably been different compared with the other samples. Fig. 4(c) shows the magnetic field dependence of the magnetization at

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Table 3 The magnetization data for the Zn0.92 Mn0.08 O nanoparticles.

Hysteresis loop Hc (Oe) Ms (emu) Mr (emu) S S* BHmax

Upward part

Downward part

Average

Parameter definition

93.827 2.752E−4 −5.454E−5 0.198 0.007 1.024E+2

−98.228 −2.785E−4 5.206E−5 0.187 0.081

96.027 2.769E−4 5.330E−5 0.193 0.044

Coercive field: field at which M/H changes sign Saturation magnetization: maximum M measured Remanent magnetization: M at H = 0 Squareness: Mr /Ms 1 − (Mr /Hc )(1/slope at Hc ) Maximum energy loss of the hysteresis loop

room temperature for Zn0.92 Mn0.08 O nanoparticles. The curve could be seen as reminiscent of uncorrected curves reported for thin film DMS, where the diamagnetic contribution of the substrate has significantly been larger than the film and is usually subtracted. In this case, there exists no substrate contribution. The ZnO background does lead to a large diamagnetic contribution. But if the entire material consists of a ferromagnetically coupled Mn-based DMS, then the expected magnetization could be significantly be found larger than the ∼10−4 emu/g (less then 0.01B Co) is observed in the Zn0.92 Mn0.08 O powders. The negative slop in the diamagnetic region is less than that anticipated from reference ZnO, indicating a paramagnetic contribution from the remaining manganese ions. The nominal atomic fractions of the dopant was x = 0.08. The M–H curves of Zn0.92 Mn0.08 O at room temperature clearly show a hysteresis loop with coercive field of 96 Oe, resulting from ferromagnetic ordering in the material. The observed ferromagnetic properties of Zn0.92 Mn0.08 O are in good agreement with the previously reported experimental results [37], but there exists a contrast with observation of a spin glass behaviour from Mn-doped ZnO films [30]. According to RKKY theory [38,39], the ferromagnetism could be attributed to an exchange interaction between local spin-polarized electrons and conductive electrons. This interaction leads to the spin polarization of conductive electrons, and also interaction between the conductive electrons. Therefore a long-range exchange interaction enables all the Mn2+ ions to have the same spin direction. As a result, the material exhibits ferromagnetism. The magnetization data of Zn0.92 Mn0.08 O nanoparticles has been presented in Table 3. 4. Conclusion In summary, it is concluded that the ZnO based diluted magnetic semiconductors (ZnO:Mn) have been studied by using an oxalate precursor decomposition method. XRD profiles have confirmed that the structures of the prepared products are in wurtzite structures without any second phases. ESR spectral analysis has enabled to verify the lattice site occupancy and also the valence state of the manganese ion in ZnO. The magnetization measurement (Fig. 4(c)) has revealed ferromagnetism in Zn0.92 Mn0.08 O sample. References [1] H. Munekata, H. Ohno, S. von Molnar, A. Segmuller, L. Chang, L. Esaki, Phys. Rev. Lett. 63 (1989) 1849.

[2] P. Fumagalli, H. Munekata, Phys. Rev. B 53 (1996) 15045. [3] R. Shioda, K. Ando, T. Hayashi, M. Tanaka, Phys. Rev. B 58 (1998) 1100. [4] S.J. Potashnik, K.C. Ku, S.H. Chun, J.J. Berry, N. Samarth, P. Schiffer, Appl. Phys. Lett. 79 (2001) 1495. [5] G.M. Schott, W. Faschinger, L.W. Molenkamp, Appl. Phys. Lett. 79 (2001) 1807. [7] K.J. Akai, Phys. Rev. Lett. 81 (1998) 3002. [8] Y.D. Park, A.T. Hanbicki, S.C. Erwin, C.S. Hellberg, J.M. Sullivan, J.E. Mattson, A. Wilson, G. Spanos, B.T. Jonker, Science 295 (2002) 651. [9] I. Malajovich, J.M. Kikkawa, D.D. Awschalom, J.J. Berry, N. Samarth, Phys. Rev. Lett. 84 (2000) 1015. [10] Y. Ohno, D.K. Young, B. Bescholen, F. Matsukura, H. Ohno, D.D. Awschalom, Nature 402 (1999) 790. [11] F.G. Monzon, H.X. Tang, M.L. Roukes, Phys. Rev. Lett. 84 (2000) 5022. [12] K. Ueda, H. Tabata, T. Kawai, Appl. Phys. Lett. 79 (2001) 988. [13] K. Rode, A. Anane, R. Mattana, J.P. Contour, O. Durand, R. LeBourgeois, J. Appl. Phys. 93 (2003) 7676. [14] J.H. Kim, J.B. Lee, H. Kim, D. Kim, Y. Ihm, W.K. Choo, IEEE Trans. Mag. 38 (2002) 2880. [15] N. Theodoropoulou, A.F. Hebard, D.P. Norton, J.D. Budai, L.A. Boatner, J.S. Lee, Z.G. Khim, Y.D. Park, M.E. Overberg, S.J. Pearton, R.G. Wilson, Solid-State Electron. 47 (2003) 2231. [16] D.P. Norton, S.J. Pearton, A.F. Hebard, N. Theodoropoulou, L.A. Boatner, Appl. Phys. Lett. 82 (2003) 239. [17] S.W. Jung, S.-J. An, G.-C. Yi, C.U. Jung, S.-I. Lee, S. Cho, Appl. Phys. Lett. 80 (2002) 4561. [18] P. Sharma, A. Gupta, K.V. Rao, F.J. Owens, R. Sharma, R. Ahuja, J.M. Osorio Guillen, B. Johansson, G.A. Gehring, Nat. Mater. 2 (2003) 673. [19] S.-J. Han, T.H. Jang, Y.B. Kim, B.-G. Park, J.-H. Park, Y.H. Jeong, Appl. Phys. Lett. 83 (2003) 920. [20] 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. [21] C. Young Mok, C. Woong Kil, K. Hyojin, K. Dojin, Y. Ihm, Appl. Phys. Lett. 80 (2002) 3358. [22] J.H. Park, M.G. Kim, H.M. Jang, S. Ryu, Y.M. Kim, Appl. Phys. Lett. 84 (2004) 1338. [23] H. Zhou, et al., Appl. Phys. Lett. 80 (2002) 210. [24] E.A. Meulenkamp, J. Phys. Chem. B 102 (1998) 5566. [25] L. Vegard, Z. Phys. 5 (1921) 17; L. Vegard, Z. Kristallogr. 67 (1928) 239. [26] A.R. Denton, N.W. Ashcroft, Phys. Rev. A 43 (1991) 3161. [27] P.B. Dorain, Phys. Rev. 112 (1958) 1058. [28] B.D. Culity, S.R. Stock, Elements of X-ray Diffraction, Prentice Hall, New Jersey, 2001. [29] K. Sato, H.K. Yoshida, Jpn. J. Appl. Phys. Part 2 39 (2000) L555. [30] T. Fukumura, Z. Jin, A. Ohtomc, H. Koinuma, M. Kawasaki, Appl. Phys. Lett. 75 (1999) 3366. [31] N. Jedrecy, H.J. von Bardeleben, Y. Zheng, J.L. Cantin, Phys. Rev. B 69 (2004) 041308. [32] J.M. Calleja, M. Cardona, Phys. Rev. B 16 (1977) 3753. [33] F. Matsukura, H. Ohno, in: K.H.J. Buschow (Ed.), T. Dietl Hand book of Magnetic Materials, vol. 14, Elsevier, Amsterdam, 2002. [34] D. Bleanny, R.S.A. Rubbins, Proc. Phys. Soc. Jpn. 77 (1961) 103. [35] K.W. Edmonds, K.Y. Wang, R.P. Campion, A.C. Neumann, N.R.S. Farley, B.L. Gallagher, C.T. Foxon, Appl. Phys. Lett 81 (2002) 4991. [36] T. Dietl, H. Ohno, F. Mastukura, J. Cibert, D. Ferrand, Science 287 (2000) 1019. [37] S.W. Jung, S.-J. An, G. Yi, C.U. Jung, S. Lee, S. Cho, Appl. Phys. Lett. 80 (2002) 4562. [38] M.A. Ruderman, C. Kittle, Phys. Rev 96 (1954) 99. [39] K. Yosida, Phys. Rev 106 (1954) 893.