Ab initio study of 59Co NMR spectra in Co2FeAl1−xSix Heusler alloys

Physica B 465 (2015) 66–70

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Ab initio study of

59

Co NMR spectra in Co2FeAl1
xSix Heusler alloys

H. Nishihara a,n, K. Sato b, H. Akai c, C. Takiguchi b, M. Geshi d, T. Kanomata e, T. Sakon a, T. Wada a a

Faculty of Science and Technology, Ryukoku University, Otsu 520-2194, Japan Division of Materials and Manufacturing Science, Graduate School of Engineering, Osaka University, Suita 565-0871, Japan c Institute for Solid State Physics, University of Tokyo, Kashiwa 277-8581, Japan d Institute for NanoScience Design, Osaka University, Toyonaka 560-8531, Japan e Research Institute for Engineering and Technology, Tohoku Gakuin University, Tagajo 985-8537, Japan b

art ic l e i nf o

a b s t r a c t

Article history: Received 3 November 2014 Received in revised form 22 February 2015 Accepted 23 February 2015 Available online 26 February 2015

Ab initio electronic structure calculation of a series of Co2FeAl1
xSix Heusler alloys has been performed, using the Korringa–Kohn–Rostoker-coherent potential approximation method to explain experimental 59 Co NMR spectra. Two prominent features are explained semi-quantitatively—a global shift of the 59Co resonance line due to alloying with Al and Si atoms in Co2FeAl1
xSix, and the effect of local disorder in creating distinct satellite lines of 59Co NMR in Co2FeAl. The importance is stressed of the positive contribution to the 59Co hyperfine field from valence electron polarization, which emerges from the halfmetallic band structure inherent in Co-based Heusler alloys. & 2015 Elsevier B.V. All rights reserved.

Keywords: Ab initio First principle 59 Co NMR Hyperfine field Heusler alloy Co2FeAl

1. Introduction Co-based Heusler alloys have received scientific and technological interest due to their potential applications as spintronics devices [1]. From electronic structure calculations, many of them have been reported to be half-metallic ferromagnets, in which the density of one of spin states at the Fermi energy vanishes, leading to 100% spin polarization at the Fermi energy [1]. Co-based Heusler alloys are especially important since they have large magnetic moments and high Curie temperatures. The experimentally observed magnetic moment per formula unit and the Curie temperature (TC) for bulk samples with L21 structure are 4.96 μB and 1170 K for Co2FeAl, and 6.0 μB and 1100 K for Co2FeSi, respectively [2,3]. There have been many reports on the nuclear magnetic resonance (NMR) spectra of thin films and bulk materials of Heusler alloys [4]. NMR studies on Co2Fe(Al, Si) Heusler alloys have mainly been performed by Inomata et al. and Wurmehl et al., independently [5–10]. They showed that the NMR spectra of 59Co in Co-based Heusler alloys are very sensitive to site disorder and are clearly distinguished among the A2, B2 and L21 structures, providing quantitative information on the level of disorder [5–10]. n

Corresponding author. Fax: þ81 77 543 7483. E-mail address: [email protected] (H. Nishihara).

http://dx.doi.org/10.1016/j.physb.2015.02.018 0921-4526/& 2015 Elsevier B.V. All rights reserved.

The NMR spectrum of 59Co from a Co2FeSi bulk sample, which has L21 structure, shows a sharp single line at 139 MHz, as is expected from the crystal structure, since all Co atoms occupy equivalent crystallographic positions [6]. On the other hand, the NMR spectrum of 59Co in thin polycrystalline films of Co2FeAl with completely disordered A2 type structure shows one broad line with a width of about 90 MHz due to the random distribution of Co, Fe, and Al atoms, while Co2FeAl with B2 structure exhibits eight resonance peaks in a frequency range of 50–300 MHz [5]. In contrast to the single line spectrum of Co2FeSi, the bulk sample of Co2FeAl, whose crystal structure is identified as L21 from the X-ray diffraction data as shown in Fig. 1, displays a distinct satellite structure around the main line at 193 MHz as shown in Fig. 2, which is taken from Ref. [6], revealing a certain degree of atomic disorder in the bulk sample [6]. The satellites extend symmetrically on both frequency sides of the main line and were interpreted to come from the Co sites whose nearest neighbors are four Fe and Al atoms (the main line), five Fe atoms and three Al atoms (the first higher satellite), three Fe atoms and five Al atoms (the first lower satellite), six Fe atoms and two Al atoms (the second higher satellite), and so on, respectively, as shown in Fig. 2 [6]. The overall spectra are well explained if 16% Fe antisites on the FeþAl sublattice is assumed, or an overlapping of 10% and 21% Fe antisites is assumed [6]. An NMR study revealed that the local structure of ascast Co2FeAl bulk samples consists of a B2 type structure with

H. Nishihara et al. / Physica B 465 (2015) 66–70

67

Fig. 1. Atomic arrangement of Co2FeAl with an L21 type Heusler structure. Fig. 3. Experimental 59Co hyperfine fields (Hexp) in Co2FeAl1
xSix, corresponding to Co atoms located in a perfect L21 environment, as well as the respective contributions from the core s shell electrons (Hcore) and the 4s valence electrons (Hvalence). Figure reprinted from Wojcik et al. [7]. Copyright 2008, American Physical Society.

Fig. 2. The 59Co NMR spectrum from bulk Co2FeAl with L21 structure. The full circles are the experimental spectrum and the continuous line is theoretical fitted spectrum, assuming L21 structure with disorder consisting of Fe and Al mixing, corresponding to a single concentration c ¼0.16 (thin line) and two concentrations c ¼0.1 and c ¼0.21 (thick line) of Fe(Al) antisites. Figure reprinted from Inomata et al. [6]. Copyright 2008, American Physical Society.

contributions of the L21 type structure of about 10% [8]. Secondary lines are also observed in the samples and attributed to the distributions of Fe and Al atoms in more distant shells [8]. For more comprehensive details, refer to the review article [4]. One aim of this report is to try to understand the satellite spectrum in Fig. 2 from ab initio (first-principles) electronic structure calculations. The analysis of such a satellite spectrum from ab initio calculations has scarcely been made to our knowledge. A systematic 59Co NMR study has been carried out for a series of Co2FeAl1
xSix polycrystalline bulk Heusler alloys [7]. It was reported that Si substitution causes a systematic shift of the entire 59 Co NMR spectrum towards lower frequencies, revealing that the

replacement of Al with Si has a global rather than local effect on the electronic structure [7]. The frequency position (hyperfine field) of the main line for the entire range of compositions decreases in a quasilinear way from 193 MHz (
19.1 T) in Co2FeAl to 139 MHz (
13.76 T) in Co2FeSi, as shown in Fig. 3, which is taken from Ref. [7]. The contributions to the hyperfine field from the core polarization effect (Hcore) were estimated in such a way that Hcore was assumed to be proportional to the magnetic moment of Co, i.e., Hcore ¼A μ (Co), and the constant A was approximated as A¼
100 kG/μB [7]. The contribution to the hyperfine field from the conduction (valence) electron polarization effect (Hvalence) was estimated as the difference between Hexp and Hcore, which are also shown in Fig. 3 [7]. It was pointed out that the experimentally observed variation in the 59Co hyperfine field on alloying mainly reflects the variation in the valence electron contribution, and the positive slope of the contribution means that the population of 4s electrons in the majority-spin band increases faster than the population in the minority-spin band, as is consistent with the theoretical scenario predicting that the electron introduced in the majority-spin band contributes to the virtual shift of the Fermi level inside the gap of the minority-spin band [7]. Second aim of this report is to understand the variation of the hyperfine fields in Fig. 3 from ab initio electronic structure calculations. There are many reports on the theoretical calculations of electronic structures of Heusler alloys including a detailed review on those of the Co2FeAl1
xSix system [1]. In the first attempt by Ishida et al. to calculate the electronic structures of some Co2based compounds such as Co2MnSn, Co2TiSi, and Co2TiAl, a minimum of the minority density of states at the Fermi energy was found but a gap was not opened [11]. The electronic structure of Co2FeAl was calculated also by Ishida et al. using the linear muffin-tin orbitals-atomic spheres approximation (LMTO-ASA) method with the local-spin density approximation [12]. The

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H. Nishihara et al. / Physica B 465 (2015) 66–70

calculated moment of 4.98 μB agrees well with the experimental value of 4.96 μB [2], and the calculated hyperfine field at the 59Co site of
128.33 kOe accounts for an experimental value of
194.0 kOe [13] semi-quantitatively. The electronic structure of Co2FeSi was calculated by Wurmehl et al. using various methods to explain the experimental magnetic moment of 6.0 μΒ [3]. According to their calculations, the LMTO method gives 5.08 μΒ . The Korringa– Kohn–Rostoker (KKR) method, within either the muffin-tin potential model or atomic sphere approximation (ASA), and with various parameterizations of the local density approximation (LDA) functional for the exchange-correlation gives calculated values of 4.88–5.67 μB, which do not correlate with the experimental values satisfactorily [3]. On the other hand, the localdensity approximation (LDA)þ U scheme predicted magnetic moment of 6.0 μB and half-metallic electronic structure [3], although this scheme is no longer a strict ab initio (first principle) calculation due to adjustable parameter U. For more comprehensive details, refer to the review article [1].

2. Calculation of the hyperfine fields and discussion Electronic structures of the Co2FeAl1
xSix system have been calculated based on the LDA of the density functional theory and the KKR Green's-function method with the coherent potential approximation (CPA) [14–17]. We used the program cpa2002v009c developed by Akai. The calculation was performed with the LDA parameterization given by Moruzzi, Janak and Williams. Experimentally determined lattice parameters, which show more or less a linear dependence on x [18], were employed for the present calculations for the concentrations x (x¼ 0, 0.3, 0.5, 0.7, and 1). The resulting spin-resolved total density of states are shown in Fig. 4 (a) together with a partial density of states of the Co s-band in Fig. 4(b). Calculated values of the total magnetic moment, the Co

atom magnetic moment, and the Co atom 4s valence electrons magnetic moment are tabulated in Table 1. Calculated magnetic moment of Fe atom was found to take nearly a constant value of 2.7 μB for the five values of x. The calculated contributions of the hyperfine fields at the 59Co site from the core polarization (Hcore) and the valence (conduction) electron polarization (Hvalence) are also tabulated, together with the calculated total hyperfine field (Hcalc_total). In the present calculations, the orbital hyperfine field and dipolar contribution have been neglected for simplicity. Small contributions of the orbital hyperfine fields have been discussed experimentally for 59Co NMR experiments in the cases of Co2TiGa and Co2VGa [19,20]. The calculated contributions of the hyperfine fields at the Co site are also shown in Fig. 5 for comparison with Fig. 3. The calculated magnetic moments change from 4.85 to 5.00 μB with increasing x in disagreement with the experimental results where the magnetic moments change from 5.0 to 6.0 μB. This disagreement is due to the limitations of LDA, which were discussed previously in Ref. [1]. LDA takes account of the exchangecorrelation energy that depends solely on the value of the electronic density at each point in space. It is well known that the calculated band gaps in semiconductors are significantly underestimated. Similarly, the half-metallicity is incomplete in Co2FeAl in LDA. In the present case of ferromagnetic Co-based Heusler alloys, the unoccupied minority d-bands are predicted at only slightly above the Fermi level and show significant amplitude for the valence minority states below the Fermi level. This causes the reduction of the total magnetic moment. The reduction of total magnetic moment becomes more pronounced with increasing x as is seen in Fig. 4. This brings the magnitude of the hyperfine field small. Actually as shown in Table 1, the present calculations explain only 72% (x ¼0) to 55% (x ¼1) of the experimental total hyperfine fields. It is quite interesting to note that the Co s-band shows

Fig. 4. (a) Spin resolved total density of states of Co2FeAl1
xSix, calculated with the KKR-CPA method based on the LDA. (b) Partial density of states of the Co-s state.

H. Nishihara et al. / Physica B 465 (2015) 66–70

69

Table 1 The total magnetic moment, the magnetic moment of the Co atom, and the magnetic moment of 4s valence electrons of the Co atom calculated with the KKR-CPA method based on the LDA are tabulated for the five values of x in Co2FeAl1
xSix. The calculated contributions of the hyperfine fields at the 59Co site from the core polarization effect (Hcore), the valence (conduction) electron polarization effect (Hvalence), and the total calculated hyperfin field (Hcalc_total) are also tabulated. Two kinds of calculated hyperfine coupling constants are shown (see text). The experimental total hyperfine field is also shown [6]. x (Si content) Total magnetic moment (μB) Magnetic moment of Co atom (μB) Magnetic moment of 4s valence electrons of Co atom (μB) Hcore(59Co) (kG) Hvalence(59Co) (kG) Hcalc_total(59Co) (kG) Hcore(59Co)/μCo Hcalc_total(59Co)/μCo Experimental Htotal(59Co) (kG) [6]

0 4.85 1.13
0.0023
127.9
10.1
138.1
113
122
193

Fig. 5. Calculated 59Co hyperfine fields at Co atoms in Co2FeAl1
xSix as a function of x using the KKR-CPA method (see the text).

half-metallic partial density of states as shown in Fig. 4(b). This feature, the complete vanishing of the minority spin density of states near the Fermi level, is also seen in Co-based Heusler alloys such as Co2MnAl, Co2MnSn, and Co2TiSn, and it seems to be a general feature of the Co s-band in the Heusler structure [21]. It is probable that the exchange-correlation effect is not important for s-bands. When Al is replaced with Si (increasing x), electrons are added and the Fermi energy shifts upward in energy relative to the whole band as seen in Fig. 4(b), where actually, the body of the majority s-band shifts down since the Fermi level is fixed in the figure. Correspondingly the majority s-band is filled further, and the positive contribution of Hvalence increases further as is seen in Table 1. The magnetic moment of the 4s valence electrons of the Co atom shown in the table is very small, but it makes an appreciable positive contribution to the hyperfine field. This is due to the fact that the hyperfine coupling constant for direct s-valence electron polarization is larger by orders of magnitude than the core polarization term which is an indirect effect as is seen in the table. Since the negative contribution from the core states is not so sensitive to x, when Al is replaced with Si, the magnitude of the total hyperfine field decreases due to the increase of the positive contribution from valence electron polarization. Therefore, the overall change in total hyperfine field is understood as a global effect of the change in the electronic structure manifested in the polarization of Co-4s states. The calculated ratio between hyperfine field and Co magnetic moment, Hcore(59Co)/μCo and Hcalc_total(59Co)/μCo, are tabulated in Table 1. The former values are nearly constant at about
110 kG/μB, and agree approximately with an empirical value of
100 kG/μB,

0.3 4.99 1.19 0.0017
132.5 21.2
111.3
112
94
174

0.5 5.05 1.21 0.0036
134.5 36.2
98.2
111
81
162

0.7 5.08 1.22 0.0049
135.1 46.8
88.3
111
72
153

1 5.00 1.18 0.0056
129.8 52.3
77.6
110
66
140

which was also used in the phenomenological analysis [7]. However, the latter values range from
122 to
66 kG/μB. This is a notable feature in Co-based Heusler alloys, where the positive contribution of the valence electron polarization has an appreciable effect on the hyperfine field due to the half-metallic nature of the s-band. One cannot deduce magnetic moments of Co atoms from total hyperfine fields obtained experimentally using the constant value of
100 kG/μB in Co-based Heusler alloys. Thus, the way of analysis reported in reference [7] is concluded to be reasonable, where the observed increase of the total magnetic moment per formula unit was assumed to be attributed to the variation of Co moment, and Hcore(59Co)/μCo was assumed to be
100 kG/μB, supporting the theoretical scenario predicting that the electron-doping to the majority-spin band leads to the virtual shift of the Fermi level inside the gap of the minority-spin band [7]. To determine the frequencies of the distinct satellite spectrum for Co2FeAl in Fig. 2, ab initio calculations were performed as follows. A supercell, shown in Fig. 1, consisting of four Co2FeAl or 16 atoms in total was considered. One Al atom, a nearest neighbor of the central Co atom, was replaced with one Fe atom, and the calculation of the hyperfine field for the first higher satellite was done using the same program, cpa2002v009c. A separate calculation was done for the first lower satellite peak, where one Fe atom as a nearest neighbor of the central Co atom was replaced with one Al atom, and so on. An experimentally obtained lattice constant of 5.73 Å was used for all supercells. The calculated magnetic moments of the Co atoms are tabulated in Table 2 for eight nearest neighbor arrangements in Co2FeAl, together with the calculated magnetic moments of the 4s valence electrons of the Co atom. The calculated contributions of the hyperfine fields from the core polarization term, Hcore, the valence electron polarization Table 2 The calculated magnetic moments of Co atoms are tabulated for various nearest neighbor arrangements in Co2FeAl, together with the calculated magnetic moments of 4s valence electrons of the Co atom, and the calculated hyperfine field contributions from the core polarization term, Hcore, the valence electron polarization term, Hvalence and the sum of these, Htotal. Magnetic Nearest neighmoment of bor arrangement of Co atom Co atom (μB)

Hcore Magnetic moment of 4s (59Co) valence elec- (kG) trons of Co atom (μB)

Hvalence (59Co) (kG)

Htotal (59Co) (kG)

1Feþ7Al 2Feþ6Al 3Feþ5Al 4Feþ4Al 5Feþ3Al 6Feþ2Al 7Feþ1Al 8Fe


0.0028
0.0031
0.0028
0.0023
0.0028
0.0032
0.0012
0.0041


20.3
19.6
15.6
10.1
12.8
15.4
15.2
21.1


53.2
89.3
115.8
138.1
161.7
182.6
199.2
218.2

0.288 0.617 0.887 1.13 1.32 1.48 1.63 1.74


32.8
69.7
100.2
127.9
148.9 -167.2
183.9
197.1

70

H. Nishihara et al. / Physica B 465 (2015) 66–70

Table 3 The calculated total hyperfine fields at 59Co sites for various nearest neighbor arrangements in Co2FeAl, and the corresponding NMR frequencies and ratios to the central frequency. The corresponding reported experimental data from Refs. [6] and [8] are also tabulated. Calculated

Experimental [6]

Experimental [8]

Nearest neighbor arrangement of Co atom

Htotal (59Co) (kG)

NMR freq. (MHz)

freq. ratio

NMR freq. (MHz)

Freq. ratio

NMR freq. (MHz)

Freq. ratio

1Feþ7Al 2Feþ6Al 3Feþ5Al 4Feþ4Al 5Feþ3Al 6Feþ2Al 7Feþ1Al 8Fe


53.2
89.3
115.8
138.1
161.7
182.6
199.2
218.2

53.4 89.8 116.4 138.8 162.6 183.6 200.2 219.4

0.39 0.65 0.84 1.00 1.17 1.32 1.44 1.58

81 122 159 193 224 251 275 –

0.42 0.63 0.82 1.00 1.16 1.30 1.42 –

– 128.6 159 192.3 223.8 248.8 268.8 287.8

– 0.67 0.83 1.00 1.16 1.30 1.40 1.50

term, Hvalence, and the sum of these, Htotal, are also shown in the table. The magnetic moment of the Co atom was found to increase with an increasing number of nearest neighbor Fe atoms. This is a major source of the change in the hyperfine fields, with the change in the nearest neighbor arrangement of Co atoms. The hyperfine coupling constant of the core polarization term is found to be about
113 kG/μB for any of the arrangements in the Table 2. The magnetic moment of the 4s valence electrons of the Co atom is very small and in the antiparallel direction to that of the Co atom, and the contribution to the total hyperfine field is only effective in the cases of nearest neighbor arrangement of the Co atom with one or two Fe atoms, where the core contribution is rather small compared to the other cases. The calculated total hyperfine fields, the corresponding NMR frequencies and the ratio of each frequency to the central frequency are tabulated in Table 3 together with the experimental values taken from Fig. 2 [6]. Independently reported experimental data from Ref. [8] are also tabulated. Although the calculated magnitude of each satellite frequency is about 73% of the experimental value, the calculated ratio of each satellite frequency to the central frequency agrees quite well with both of the reported experimental ratios. In summary, ab initio calculations of the hyperfine fields of 59Co in the Co2FeAl1
xSix system, with the KKR-CPA method based on the LDA, are found to explain both of the prominent features semiquantitatively: the global shift of the 59Co resonance line due to alloying with Al and Si atoms, and the effect of very local disorder in creating distinct satellite lines of 59Co NMR in Co2FeAl.

Acknowledgments One of the authors (H. N.) thanks Dr. T. Fukushima, Dr. M. Inukai, Mr. S. Yamamoto, Dr. T. Doi, and Mr. K. Ito for helpful discussions at the 25th Computational Material Design Workshop. This study was partly supported by a Grant-in-Aid for Scientific Research (C) (Grant no. 21560693) from the Japan Society for the

Promotion of Science (JSPS) of the Ministry of Education, Culture, Sports, Science and Technology, Japan. Also thanks to Enago for the English language review.

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