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Graphene (Gr) is one of the most outstanding materials for room temperature (RT) spintronics due to its unique and intrinsically tunable electronic properties [1,2]. Some important effects such as high spin injection efficiency, antiferromagnetic coupling, Rashba, spin filtering, TMR or enhanced PMA have been found when Gr is coupled with ferromagnetic materials. Its electronic properties can be tuned by proximity with heavier metallic atoms, which may allow for the technological development of more efficient devices based on spin-orbit effects. However, the development of Gr-based hybrid structures requires the capability of engineering high-quality interfaces, as it has a direct impact in the functional properties of the system [3]. On this regard, metal intercalation has been widely used for the fabrication of functional Gr/FM/HM heterostructures. Gr promotes a layer-by-layer growth of the metal at the interface with a heavy metal substrate, resulting in high quality epitaxial interfaces. Moreover, Gr protects the intercalated material from air exposition avoiding its oxidation and preserving its physical properties. It has been recently demonstrated by the authors that if the intercalated species is a FM, one can prepared highly perfect and flat hybrid Gr/FM/HM epitaxial heterostructures onto insulating oxides with enhanced perpendicular magnetic anisotropy (PMA), large DMI and stable Néel-type chiral magnetic nanometer sized objects [4].&lt;br/&gt;In the present work we report on the fabrication of high quality epitaxial Gr/Fe/Pt asymmetric magnetic structures with extremely abrupt interfaces. Samples were fabricated entirely in UHV following a methodology previously stablished [5]. Thermal intercalation of Fe in Gr/Pt(111) interfaces is studied in detail by means of X-ray photoelectron spectroscopy, observing a shift of the C1s core level main contribution to higher binding energies due to the transition from n-doped to p-doped graphene due to graphene-iron interaction. High surface and interface quality is demonstrated by means of HR-TEEM, LEED and XRR. Atomic resolution images prove an epitaxial and conmensurate growth of Fe on Pt(111) with a characteristic ABC stacking of the fcc structure. EELS elemental mapping of the heterostructures confirms the good chemical differentiation of all the layers in the system. Magnetic characterization was performed by MOKE at ambient conditions varying Fe thickness in a range between 1 to 4 MLs. Our results show the presence of PMA in the system for 1 ML of Fe, with a complete transition to in plane magnetization for thicknesses above 3 MLs.

**References:**

[1] A. H. Castro Neto et al., &lt;i&gt;Rev. Mod. Phys&lt;/i&gt;. 81, 109-162 (2009)&lt;br/&gt;[2] W. Han et al., &lt;i&gt;Nature Nanotech&lt;/i&gt; 9, 794–807 (2014)&lt;br/&gt;[3] W. J. Wells et al., &lt;i&gt;Phys. Rev. B &lt;/i&gt;95, 054428 (2017)&lt;br/&gt;[4] F. Ajejas et al., &lt;i&gt;Nano Lett.&lt;/i&gt; 18, 9, 5364-5372 (2018)&lt;br/&gt;[5] F. Ajejas et al., &lt;i&gt;ACS Appl. Mater. Interfaces.&lt;/i&gt; 12, 4088-4096 (2020)

**Images:**





INTERMAG 2021

 Tuning work function in Graphene by thermally assisted metal intercalation

The 65th Annual Conference on Magnetism and Magnetic Materials (MMM 2020) was held in a virtual form. The COVID-19 pandemic, with its associated uncertainty about international travel and the ability to gather in large groups, meant that the Steering Committee decided to cancel the face-to-face meeting that had been planned in Palm Beach, Florida November 2-6, 2020.

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Graphene (Gr) is one of the most outstanding materials for room temperature (RT) spintronics due to its unique and intrinsically tunable electronic properties [1,2]. Some important effects such as high spin injection efficiency, antiferromagnetic coupling, Rashba, spin filtering, TMR or enhanced PMA have been found when Gr is coupled with ferromagnetic materials. Its electronic properties can be tuned by proximity with heavier metallic atoms, which may allow for the technological development of more efficient devices based on spin-orbit effects. However, the development of Gr-based hybrid structures requires the capability of engineering high-quality interfaces, as it has a direct impact in the functional properties of the system [3]. On this regard, metal intercalation has been widely used for the fabrication of functional Gr/FM/HM heterostructures. Gr promotes a layer-by-layer growth of the metal at the interface with a heavy metal substrate, resulting in high quality epitaxial interfaces. Moreover, Gr protects the intercalated material from air exposition avoiding its oxidation and preserving its physical properties. It has been recently demonstrated by the authors that if the intercalated species is a FM, one can prepared highly perfect and flat hybrid Gr/FM/HM epitaxial heterostructures onto insulating oxides with enhanced perpendicular magnetic anisotropy (PMA), large DMI and stable Néel-type chiral magnetic nanometer sized objects [4]. In the present work we report on the fabrication of high quality epitaxial Gr/Fe/Pt asymmetric magnetic structures with extremely abrupt interfaces. Samples were fabricated entirely in UHV following a methodology previously stablished [5]. Thermal intercalation of Fe in Gr/Pt(111) interfaces is studied in detail by means of X-ray photoelectron spectroscopy, observing a shift of the C1s core level main contribution to higher binding energies due to the transition from n-doped to p-doped graphene due to graphene-iron interaction. High surface and interface quality is demonstrated by means of HR-TEEM, LEED and XRR. Atomic resolution images prove an epitaxial and conmensurate growth of Fe on Pt(111) with a characteristic ABC stacking of the fcc structure. EELS elemental mapping of the heterostructures confirms the good chemical differentiation of all the layers in the system. Magnetic characterization was performed by MOKE at ambient conditions varying Fe thickness in a range between 1 to 4 MLs. Our results show the presence of PMA in the system for 1 ML of Fe, with a complete transition to in plane magnetization for thicknesses above 3 MLs.

**References:**

[1] A. H. Castro Neto et al., Rev. Mod. Phys. 81, 109-162 (2009) [2] W. Han et al., Nature Nanotech 9, 794–807 (2014) [3] W. J. Wells et al., Phys. Rev. B 95, 054428 (2017) [4] F. Ajejas et al., Nano Lett. 18, 9, 5364-5372 (2018) [5] F. Ajejas et al., ACS Appl. Mater. Interfaces. 12, 4088-4096 (2020)

**Images:**

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Over the past few years, spintronic devices have attracted considerable attention due to their potential application compared to the classical electronics. Recently, a new concept of racetrack memory (RM) is developed, where domain walls (DWS) are replaced by magnetic chiral spin textures (skyrmions) [1] stabilized in the presence of the interfacial Dzyaloshinskii-Moriya interaction (iDMI). Their small size, high speed, and low energy consumption make skyrmions good candidates for future spintronic data storage. Skyrmion-RM will would allow to considerably increasing storage densities and information transfer speed [1, 2]. Therefore such devices performance depends significantly on the proper modulation of the Nonmagnetic/ Ferromagnetic (FM) interface and interfacial associated phenomena. In addition to the Heavy-metal (HM)/ (FM) interfaces and related spin-orbit coupling (SOC) interaction, it has recently been demonstrated the presence of the perpendicular magnetic anisotropy (PMA) and the iDMI in the FM/ Metal Oxide (Mox) interfaces. The PMA in such interface is due to the hybridization between the ferromagnetic metal and oxygen ions [3]. Recently, density functional theory has predicted the controlling of the iDMI sign and magnitude by changing the oxygen coverage of the FM layer [4]. However, knowing that these effects are dependent on the interface configuration, the study of the iDMI in the FM/MOx is still lacking. An efficient way to determine the role and the contribution of such interface is to investigate a FM/MOx system with or without a HM underlayer. Thus, in this work we propose the investigation of the PMA and, the iDMI as a function of the annealing temperature, the PtOx thickness and the oxidation state in CoFeB/PtOx interfaces. We expect to obtain overoxidation state for the thinner Pt films, where oxygen penetrates the underlying magnetic layer, whereas underoxidazied and optimally oxidized states are obtained for thicker Pt layers. Two categories of samples were grown by DC magnetron sputtering on thermally oxidized silicon, with or without Ta (3 nm) buffer layer. In the first category, the Co8Fe72B20 (CFB) thickness was varied up to 10 nm (2nm≦ tCFB≦10 nm) and capped by TaOx (0.8 nm). In the second category, CFB thickness was kept at 1.5 nm and was capped by variable thicknesses of PtOx (0.7nm≦ tPt≦1.6 nm). Both as grown and 225 °C annealed are studied for this second category. For this experimental investigation, Vibrating sample magnetometer (VSM), Brillouin light scattering (BLS) in Damon-Esbach (DE) configuration and Microstrip line ferromagnetic resonance (MS-FMR) were combined. The VSM was used to investigate the static magnetic properties (magnetization at saturation (Ms) and magnetic dead layer thickness (td)). The FMR was used for the determination of the gyromagnetic ratio, and for the investigation of the PMA in the thicker films. The BLS was employed for the study of the PMA (thinner films) and the iDMI. For the PMA, the spectra were measured at spin wave (SW) vector ( ksw= 8.08 µm−1) under variable in-plane applied magnetic fields, while for the iDMI measurements, BLS spectra were measured as a function of the ksw under in-plane saturating magnetic field. VSM measurement revealed that td is approximately the same for the CFB/TaOx (td≈ 0.7 nm) and CFB/PtOx (td≈ 1 nm) systems. For the CFB/PtOx stackcs we notice slightly higher value for the samples with Ta buffer layer due to the supplement of intermixing at the bottom interface. The larger value of the td (≈ 1 nm) obtained for CFB/PtOx compared to CFB/TaOx samples may be due to the fact that the Pt does not oxidize, the oxygen penetrates more into the CFB films thus increasing the thickness of the magnetic dead layer. Ms is also approximately the same for all the samples (around 1510 emu/cm3) which is in good agreement for systems of CFB rich on Fe [5]. The PMA investigation in the CFB/TaOx samples has shown the degradation of the interface anisotropy by inserting a Ta buffer layer. Indeed, the surface anisotropy constant (Ks) in the Ta/CFB/TaOx is twice weaker for the samples without Ta buffer layer confirming the degradation of the interface anisotropy by the Ta buffer layer. The PMA and iDMI measurements as a function of the Pt thickness have shown that the effective magnetization (Meff) and the effective iDMI constant (Deff) have the same trend for all the systems. Deff and Meff increase with increasing tPt thus allowing characterizing the strength of the iDMI and PMA for the CFB/PtOx interface. In the as deposited CFB/PtOx films, the thinnest Pt (tPt = 0.7 nm) sample is perpendicularly magnetized. The increasing of Meff as Pt thickness increases suggested that the oxidation of CoFeB decreases. The experimental data of Meff and Deff were then fitted using exponential grow function, to extract a decay thickness (λ). It is found that, λ for the Meff is higher compared to that of Deff for as grown CFB/PtOx stacks, confirming that the iDMI is slightly more localized at the first monolayers at the interface compared to the PMA. Measurements of spin pumping induced damping have also been performed by FMR and BLS will also be presented.

**References:**

[1] R. Tomasello, E. Martine., R. Zivieri et al. . Sci Rep 4, 6784, (2014) [2] W. Kang, X. Chen, D. Zhu et al., IEEE 7th Non-Volatile Memory Systems and Applications Symposium (NVMSA), Hakodate, pp. 7-12, (2018) [3] S. Monso, B. Rodmacq, S. Auffret, et al. Applied Physics Letters, 80, no 22, pp 4157, (2002) [4] A. Belabbes, G. Bihlmayer, S. Blügel, et al. Scientific Reports 6, 24634, (2016). [5] T. Ogasawara, M. Oogane, M. Al-Mahdawi et al, AIP Adv. 9, 125053, (2019).

**Images:**

 Interfacial Dzyaloshinskii Moriya interaction and perpendicular magnetic anisotropy in CoFeB/PtOx based systems

Cobalt ferrite (CoFe2O4, CFO) is a material used for technological applications such as permanent magnets, spintronic devices and recording media. This is due to its high magnetocrystalline anisotropy, large saturation magnetization in comparison to other ferrites, elevated Curie temperature and large magnetostrictive constant. Structurally, the ideal CFO corresponds to an inverse spinel (AB2O4), i.e with all the cobalt cations (Co2+) and half of the iron ions (Fe3+) located in octahedral sites (B), and the other half of the Fe3+ occupying tetrahedral sites (A). However, in the case of thin films, depending on the growth conditions and subsequent annealing treatments, a fraction of the cobalt cations can be forced to move to tetrahedral sites. The subject of cationic distribution in cobalt ferrite thin films has been already addressed in the past, with Mössbauer spectroscopy playing a fundamental role. In most cases, however, the films have been grown on oxide substrates whilst films grown on metallic substrates have been scarcely studied. In this work, we have studied the properties of ultrathin CoFe2O4 films grown by oxygen assisted molecular beam epitaxy on a Pt(111) single crystal. The deposition was performed at 523 K temperature, and the resulting film was subsequently annealed at intermediate temperatures (up to 773 K) in order to prevent the formation of multiphasic films, island growth or dewetting phenomena. Samples were grown from a few nanometers thickness to 20 nm. We characterized their composition and structural properties after each step of the growth process by Conversion Electron Mössbauer and Auger electron spectroscopies (CEMS and AES, respectively), low energy electron diffraction (LEED), and scanning tunneling microscopy (STM). An important particularity of this work was that both the growth and the characterization of the films were performed in-situ, i.e. in the same ultra high vacuum chamber, without exposure of the samples to the laboratory atmosphere in order to avoid any chemical transformations resulting from such exposure.

**References:**

V.A.M. Brabers, Handbook of Magnetic Materials, Vol. 8, pp.189–324 (1995) J. Smit and H. Wijn, Philips’ technical library, Vol. 278, pp. 136-172 (1959) M. De Santis et al, Acta Cryst B, Vol. 75, pp. 8-17 (2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/b6c2363e9d22f051bc8da9b2820f1ee6.jpg)
Fig 1. Mössbauer spectra obtained for the 5 nm cobalt ferrite thin film for the different stages: a) as grown measured at room temperature, b) and c) annealed to 673 K and 773 K measured at room temperature and 125 K, respectively.

 Ultrathin CoFe2O4 films grown by molecular beam epitaxy on Pt(111)

The properties of superconductor/ferromagnet multilayers and junctions have been intensively studied from the standpoint of the interplay between superconductivity and ferromagnetism. Though in a special case superconductivity and ferromagnetism can coexist and exhibit novel phenomena such as π state in superconductor/ferromagnet/superconductor junctions [1], in general, the superconducting order parameter is expected to suppress in the superconducting layer but to penetrate into the ferromagnetic layer. A number of the studies have been carried out with ferromagnetic component consisting of 3d transition metals like Fe, Co and Ni [2]. In contrast, only few reports exist for epitaxial NbN/Heusler alloy layered films and their devices so far; for example, giant coercivity enhancement in NbN/Co2FeSi bilayers [3], efficient injection of pure spin currents in NbN/Co2MnSi devices [4], superconductivity in NbN/Co2(Fe,Mn)Si/Au trilayers [5,6], and spin polarization measurements of Co2FeSi by the epitaxial contact Andreev reflection spectroscopy [7]. Therefore, we focus on superconducting properties of epitaxial films of a superconducting NbN layer attached to a highly spin polarized Heusler alloy Co2FeSi layer, where the superconductivity of the NbN layer penetrates into the ferromagnetic Co2FeSi layer. We investigate how the Co2FeSi layer with the high spin polarization affects the superconductivity of the NbN layer in epitaxial NbN/Co2FeSi/Au films through their interface. We have fabricated NbN/Co2FeSi/Au epitaxial films on an MgO-substrate by an ultrahigh-vacuum (UHV)-compatible magnetron sputtering system. The thickness of the CFMS film was 5 nm while the thickness of the NbN film was fixed to 100 nm. Considering the NbN film was thicker enough than its coherence length ξNbN(0) at 0 K, the superconductivity of the layered film should be in the three-dimensional (3D) regime, due to satisfying dNbN &gt;&gt; ξNbN(0). Superconducting properties of the epitaxial films were evaluated by resistivity measurements with the standard four-terminal technique under both parallel and perpendicular magnetic fields up to 17 T. The superconducting transition temperature Tc of NbN/Co2FeSi/Au trilayer films was determined to be 16.1 K in the absence of magnetic field. Temperature dependence of the resistivity ρ(T) was measured in both magnetic fields parallel and perpendicular to the surface of NbN/Co2FeSi/Au trilayer films. Figure 1 shows the field dependence of the pinning potential U0(H) for both applied magnetic field directions deduced from the Arrhenius plots for the ρ(T,H) curves. As shown in Fig. 1, U0(H) was well described by H−α(1−H/H*)β above 5 T by the similar model with the exponents in the field dependence of the pinning force density, where H* is a characteristic field representing the irreversibility field at 0 K. Temperature dependence of the upper critical field μ0Hc2(T) was evaluated from the ρ(T) measurements. Figure 2 represents the upper critical fields μ0Hc2||(T) and μ0Hc2⊥(T) of a NbN/Co2FeSi/Au trilayer film in two magnetic field directions applied parallel and perpendicular to the surface of the trilayer From Fig. 2, the upper critical field Hc2(0) at 0 K was also deduced to be μ0Hc2||(0) = 23.2 T for the parallel magnetic field and μ0Hc2⊥(0) = 16.2 T for the perpendicular magnetic field, respectively. The experimental results under magnetic fields revealed the superconductivity of the NbN layer was affected by the interplay between the superconducting NbN layer and the highly spin polarized Co2FeSi layer.

**References:**

[1] T. Kontos, M. Aprili, J. Lesueur, F. Genêt, B. Stephanidis, R. Boursier, Phys. Rev. Lett. 89 (2002) 137007. [2] L. Lazar, K. Westerholt, H. Zabel, L. R. Tagirov, Y. V. Goryunov, N. N. Garif'yanov, I. A. Garifullin, Phys. Rev. B 61 (2000) 3711. [3] A. Gupta, G. Singh, D. Kumar, H. Kishan, and R. C. Budhani, Appl. Phys. Lett. 103 (2013) 182602. [4] D. Kumar, P. C. Joshi, Z. Hossain, and R. C. Budhani, Appl. Phys. Lett. 102 (2013) 112409. [5] I. Shigeta, T. Kubota, K. Makise, S. Kimura, S. Awaji, B. Shinozaki, K. Koyama, K. Takanashi, and M. Hiroi, IEEE Magn. Lett. 8 (2017) 3305605. [6] I. Shigeta, T. Kubota, Y. Sakuraba, S. Kimura, S. Awaji, K. Takanashi, and M. Hiroi, Physica B 536 (2018) 310. [7] I. Shigeta, T. Kubota, Y. Sakuraba, C. G. Molenaar, J. N. Beukers, S. Kimura, A. A. Golubov, A. Brinkman, S. Awaji, K. Takanashi, and Masahiko Hiroi, Appl. Phys. Lett. 112 (2018) 072402.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/33cb1ef7173a0adf01a17dd4170bd8e5.jpg)
Fig. 1. Pinning potential U0(H) as a function of magnetic field H. The blue circles are U0(H) parallel magnetic fields H|| to the surface for the NbN/Co2FeSi/Au trilayer film and the red squares are perpendicular magnetic fields H⊥. Inset: Relationship between ln ρ0f and U0 for both magnetic fields H|| and H⊥.

![](https://assets.underline.io/uploads/markdown_image/1/image/508049c8022712d9aedc6040944a89ff.jpg)
Fig. 2. Temperature dependence of the upper critical field Hc2(T) for both directions deduced from ρ(T) measurements at various magnetic fields.

 Superconducting properties of epitaxial films of superconducting NbN and highly spin polarized Co2FeSi under high magnetic fields

Antiferromagnetic coupling between magnetic films has long been studied for the design and tuning of spin structures, and is a core issue in spintronics research. Such a coupling has been widely reported in metallic multilayers, but the effect is seldom seen in the more complex multilayer film interfaces [1]. Here we studied the antiferromagnetic coupling at the interface between terbium iron garnet (Tb3Fe5O12, TbIG) and yttrium iron garnet (Y3Fe5O12, YIG) thin films, being manifested in form of an inverted hysteresis loop at low temperatures. TbIG (7 nm)/YIG (23 nm) was deposited on Y3Al5O12 (YAG) substrates with (110) orientation by pulsed laser deposition at 437 K under 100 mTorr oxygen pressure. As a control, a sample with 5 nm Gd3Ga5O12 (GGG) spacer between the two iron garnet layers was also prepared. The crystal structure of the films was characterized by a high-resolution X-ray diffractometer (HRXRD). The surface morphology of samples was characterized by atomic force microscopy (AFM). The microstructures and thickness of the samples were examined by high-resolution transmission electron microscopy (HRTEM). For transport measurements, Pt (5 nm) Hall bar patterns were deposited on the garnet samples by rf sputtering. Anomalous-Hall effect (AHE) of samples was measured by a physical property measurement system (PPMS). The magnetic properties of the samples were measured by the vibrating sample magnetometer (VSM) option of the PPMS. Fig. 1(a) shows the HRXRD pattern of TbIG (7 nm)/YIG (23 nm), demonstrating epitaxial growth of the films on YAG with strong (110) orientation (rocking curve with a full width at half maximum (FWHM) of 0.064°, Fig. 1(b)). AFM scan indicates a small root mean square roughness of 0.38 nm, with clear atomic terraces. Fig. 1(c) and (d) show the in-plane magnetic hysteresis loops of single layer YIG (30 nm) and TbIG (30 nm) measured at 10 K, respectively; both films were deposited on YAG (110) substrates under the aforementioned conditions. Due to the strong contribution from the Tb at low temperatures, the TbIG monolayer film exhibited much stronger magnetization than YIG of the same thickness did. A strong magnetocrystalline anisotropy was observed in the TbIG monolayer as evidenced by the high coercivity (Hc-TbIG = 5.5 kOe; Hc-YIG = 0.19 kOe) [2]. As shown in Fig. 2(a), the hysteresis loop of the TbIG (7 nm)/YIG (23 nm) sample measured at 10 K shows an obvious soft/hard magnetic coupling phenomenon, supported by the appearance of double coercivity behavior. The switching at the low field was mainly attributed to YIG, while the reversal at the high field was attributed to TbIG. The most characteristic feature of the hysteresis curve was the inversion under the low magnetic field (Inset of Fig. 2(a)): when the external field decreased from the positive saturation, the YIG layer first reversed its magnetization at a small positive field, while the magnetization of TbIG remained undisturbed. This clearly indicated the antiferromagnetic coupling between the TbIG/YIG at the interface [3]. In the case of the control sample, when a 5 nm GGG spacer is present at the TbIG/YIG interface, one could only observe the normal magnetic hysteresis loop as shown in Fig. 2(b). The inverted loop at the low field disappeared but the double coercivity could still be clearly observed, indicating the inverted loop at the low field was caused by the antiferromagnetic coupling at the interface of two different ferrimagnets. In summary, the antiferromagnetic coupling effect has been observed between two neighboring ferrimagnets, manifested in the form of an inverted hysteresis loop at the low magnetic field. Temperature-dependence of such an antiferromagnetic coupling effect and the anomalous-Hall effect of samples would be conducted to further elucidate the nature of the coupling effect. Financial support by the RGC, HKSAR (PolyU 15302320) is acknowledged.

**References:**

[1] T. Maity, D. Kepaptsoglou and S. Roy, "Observation of complete inversion of the hysteresis loop in a bimodal magnetic thin film," Physical Review B., 95(10):100401, 2017. [2] R. F. Pearson, "Magnetocrystalline Anisotropy of Rare-Earth Iron Garnets," Journal of Applied Physics., 33(3):1236-1242, 1962. [3] H. Zhang, J. Zhang and J. R. Sun, "Antiferromagnetic interlayer coupling of (111)-oriented La0.67Sr0.33MnO3/SrRuO3 superlattices," Chinese Physics B., 28(3):037501, 2019.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/1d530bd49dfedcea853b1f515b39583c.jpg)
Fig.1. (a) θ/2θ scan and (b) rocking curve of TbIG (7 nm)/YIG (23 nm)/YAG (110) sample. Inset of (a) shows the AFM image (scan height = 10 nm). In-plane magnetic hysteresis loops of YIG (30 nm) and TbIG (30 nm) single layer films at 10 K are shown in (c) and (d), respectively.

![](https://assets.underline.io/uploads/markdown_image/1/image/931b651b61d4958badb65d7240caf8fe.jpg)
Fig 2. In-plane magnetic hysteresis loops of (a) TbIG (7 nm)/YIG (23 nm) and (b) TbIG (7 nm)/GGG (5 nm)/YIG (23 nm) samples at 10 K. The insets in each figure show the corresponding enlarged images at low field. The arrows indicate the field sweep directions.

 Observation of Antiferromagnetic Coupling between Ferrimagnetic Garnet Thin Films

As one of the methods to induce uniaxial magnetic anisotropy in magnetic films, the method of “periodically patterning of films” has attracted more and more research interest because of its promising application in magnetic electronic devices. It has been confirmed that in-plane uniaxial anisotropy of magnetic films can be induced by substrate with ripple patterns，and the direction of easy axis of induced anisotropy is parallel to the ripple direction [1]. Ki in Ref. 2 produced strong in-plane uniaxial anisotropy in NiFe films by annealing the sapphire substrate to form substrate of periodical ripple patterns; the anisotropy in samples is from the dipole effect by surface magnetic charge and shape anisotropy by rippled surface morphology [2]. However, the relationship between annealing-time dependent faceting and coarsening mechanisms of substrate on permeability spectrum of films has not been clarified. In our work, we fixed the annealing temperature at 1400 degrees centigrade and changed the annealing time of substrate to prepare rippled sapphire substrates with different periodicity. Then, we deposited Ta (5 nm)/NiFe (50 nm)/Ta (2 nm) three-layer films on these substrates to investigate the annealing-time dependent faceting and coarsening mechanism of substrate on permeability spectrum of film. The surface morphology of substrates was characterized by atomic-force microscopy, see Fig. 1. The effective modulation of substrate ripple is confirmed. Fig. 2 (a)-(f) show the permeability spectra of real parts μ' and imaginary parts μ'' at no external field for films deposited on substrates annealed for 0-10 h. The results show both μ' and μ'' of NiFe films that were deposited on substrates with periodical ripples are smaller than the film on substrate not annealed. The magnitude of permeability can be modulated by annealing time of substrate and it is related to the number of ripples per unit area and the geometric size of the ripples. The resonance frequency of films on substrates with periodical ripples is higher than that of film on substrate not annealed because of the in-plane uniaxial magnetic anisotropy induced by rippled patterns. This work provides an optimized annealing time to obtain high natural resonance frequency and a method to design permeability of magnetic films. This work would be helpful for film-based magnetic devices on ripple substrate.

**References:**

[1] Sarathlal K V, Kumar D, Gupta A, Appl. Phys. Lett. 98, 123111 (2011). [2] Ki S and Dho J, Appl. Phys. Lett. 106, 212404 (2015).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/10f0e07017bd484924898841ec835ef7.jpg)
Fig. 1 Dependence of wavelength and height of substrate ripples on different annealing time. The inset maps are surface morphologies of these annealed substrates characterized by atomic-force microscopy.

![](https://assets.underline.io/uploads/markdown_image/1/image/96353c611ffb4f505e21b8cb9c14c705.jpg)
Fig. 2 Permeability spectra of real parts μ' and imaginary parts μ'' at no external field for films deposited on substrates annealed for different time: (a) 0 h, (b) 2 h, (c) 4 h, (d) 6 h, (e) 8 h, (f) 10 h.

 Modulated in-plane uniaxial magnetic anisotropy and permeability spectrum of NiFe films by sapphire substrates with periodical ripples

The exchange bias effect, which manifests itself as a shift of the hysteresis loop of a ferromagnet when it is adjacent to an antiferromagnet, is a key feature used in such technological spheres as magnetic recording and storage as well as magnetic sensorics [1,2]. The exchange bias field Hex – an amount by which the loop is shifted along the magnetic field axis – as well as the coercivity Hc are very sensitive to the temperature. Changes in both of these characteristics are directly connected with changes of magnetic properties of the layers, mainly, the anisotropies of the layers. Directly determining this parameter for an antiferromagnet is often a demanding task, especially dealing with a nanostructure. This task, however, is not only of fundamental, but of applied interest too. By systemizing a collection of data on currently used antiferromagnets it is then possible to propose brand new ones, suitable for technological applications. The industrial standards currently are Pt- and Ir-based antiferromagnets, in which the presence of precious metals substantially affects the overall cost of manufacturing. Recently scientists have been studying nitrides, oxides and Heusler-type compounds as affordable alternatives, but those come with their own limitations, such as relatively low Néel temperatures or high interdiffusion tendency. In this regard use of machine learning for prediction of possible new antiferromagnets seems rather promising, although it requires a dataset on existing antiferromagnets. For example, one of the most crucial properties of any magnetic material, and especially of an antiferromagnet is its anisotropy constant [3]. One way of obtaining this data is by using different measurement protocols. For instance, the York protocol [3,4] has proven itself as a useful tool that by subsequent measurements of the hysteresis loops (Fig. 1) of an ferromagnet/antiferromagnet bilayer estimate the anisotropy constant of the antiferromagnet. All the measurements are conducted at 100 K, but between them the sample is activated: it is maintained at subsequently raising temperatures Tact during the activation time t in a field of the sign opposite to that in which the sample was set initially. This approach is based on the fact that magnetron-sputtered films, such as those considered in this work, have a grain-size distribution of crystallites. Due to this distribution at any finite temperature the whole antiferromagnetic layer divides into three categories: 1) stable grains, which pin the ferromagnet’s magnetization thus contributing to Hex; 2) unstable grains, which irreversibly follow the ferromagnet’s magnetization thus contributing to Hc; 3) paramagnetic and superparamagnetic grains, which do not contribute to the hysteresis. This segregation is due to the fact that the energy of a grain depends on its volume and when it equals the thermal energy, a grain becomes unstable. The equation is as follows: KAF V = kB T (1), where KAF is the anisotropy constant of an antiferromagnet, V is the volume of a grain, kB is the Boltzmann’s constant and T is temperature. Through a series of field-cooling procedures alternating with hysteresis loop measurements it is possible to estimate the median blocking temperature &lt;TB&gt; at which exactly one half of the type 1 grains pin the magnetization in one direction while the other half does so in the opposite direction so that the net shift of the loop is zero (Fig. 2). The value of the anisotropy constant KAF at this particular temperature can be calculated by using the following equation: KAF (&lt;TB&gt;) = (KB &lt;TB&gt; ln (t f0))/Vm (2), where t is the time for which the setting field was applied, set to be 30 min in our experiment, f0 is switching attempt frequency from the Néel-Arrhenius law taken to be 2.1 1012 s-1[5], Vm is the median volume of the grains, which was estimated by using the X-ray diffractometry and electron microscopy. In this work by using the York protocol and employing a vibrating sample magnetometer with a temperature chamber we estimate the anisotropy constants of the antiferromagnetic Ni-Mn in a series of magnetron-sputtered Fe-Ni/Ni-Mn films of different thickness of the Ni-Mn layer. Ni-Mn is a metallic antiferromagnet, systems based on which are capable of sustaining exchange bias up to temperatures as high as 700 K [6]. Using this approach, we estimated that a 20-nm thick Ni-Mn has KAF (320 K) = 2.1 106 erg/cc. Our results can help to form a dataset on the anisotropy constants of antiferromagnets and can also be used to study the finite-size scaling effect in Ni-Mn. This work was financially supported by the Russian Science Foundation (RSC), project No. 18-72-10044.

**References:**

F. Radu, H. Zabel, Magnetic heterostructures, Springer, 97 (2008) J. Nogués, J. Sort, V. Vanglais, et.al., Phys. Rep., Vol. 422, p.65 (2005) K. O’Grady, J. Sinclair, K. Elphick, et. al., J. Appl. Phys., Vol. 128, p.040901 (2020) K. O’Grady, L.E. Fernandez-Outon, G. Vallejo-Fernandez, JMMM, Vol. 322, p.883 (2010) S. Jenkins, R.W. Chantrell, T.J. Clemmer, et. al., Phys. Rev. B, Vol. 100, p.220405(R) (2019) V.O. Vas'kovskiy et al., J. Alloys Compd., Vol. 777, p.264 (2019)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/c0d8de8aac5aab99d9374a4497ab0d7a.jpg)
Fig. 1. Hysteresis loops of a Fe20Ni80 (40 nm)/Ni30Mn70 (20nm) film obtained at 100 K following the York protocol after being set at an activation temperature Tact.

![](https://assets.underline.io/uploads/markdown_image/1/image/dc6843a0d73b5424ac6cbd49931ce501.jpg)
Fig. 2. Measurement of the blocking temperature for a Fe20Ni80 (40 nm)/Ni30Mn70 (20nm) film. The arrow denotes the median blocking temperature <TB>.

 Estimating the Anisotropy Constant in an Antiferromagnet Through Exchange Bias in Polycrystalline Ni-Mn/Fe-Ni Films

Ferrimagnetic films of rare earth-transition metal (RE-TM) alloys are successfully used as an effective source of internal magnetic bias in exchange-coupled magnetically hard/soft bilayers, including those which demonstrate a hysteresis-free magnetization process. The last feature is in demand for many applications, for example, magnetic sensors [1]. It is known that the magnetic properties of RE-TM films are largely determined by their chemical composition. In particular, amorphous Tb-Co films with the Tb content in the range of 15-40 at. % can display a perpendicular magnetic anisotropy [2]. Within a certain temperature range, the shape anisotropy becomes dominant, and the magnetization orients in the film plane. Thus, the spontaneous spin reorientation transition takes place [3]. Therefore, in these exchange-spring systems, the magnetizations of adjacent soft magnetic and hard magnetic layers can lie in the plane of the film, be perpendicular to the plane, or be orthogonal to each other. Models of the magnetization reversal processes for both planar and perpendicular exchange-spring systems are successfully developed [1,4,5]. At the same time, the reason for the onset of exchange interaction between layers, the magnetizations of which are orthogonal to each other, is not entirely clear. Several models have been proposed to explain this phenomenon. Within one of them, the RE-TM layer is considered as a heterophase system, the magnetizations of different phases are oriented parallel or perpendicular to the film plane [6]. The second model assumes that there is a transition region in the RE-TM layer, within which a gradual reversal of spins occurs from an orientation parallel to the interface of the layers to a perpendicular one [7]. The third model assumes that the exchange bias effect originates from the chiral spin texture of the interfacial domain wall. The creation of the chiral interfacial domain wall is a consequence of the interplay between Dzyaloshinskii-Moriya-interaction in Dy-Co and ferromagnetic exchange coupling between the Ni-Fe and Dy-Co layers [8]. Investigation of changes in the magnetic properties of hard/soft bilayers near the spin reorientation transition of ferrimagnetic hard layer can provide additional information on the exchange bias mechanism in the systems with orthogonal orientation of layer magnetizations. This work presents the results of a study of the magnetic properties of exchange-coupled Tb-Co/FeNi films with different composition of Tb-Co layer in a wide temperature range. Ta(5 nm)/Tb-Co(20 nm)/FeNi(40 nm)/Ta(5 nm) multilayered films were prepared by magnetron sputtering. The films were deposited on Corning glass substrates in the presence of a constant magnetic field of 250 Oe oriented in the plane of the substrate. Magnetic measurements of the multilayered films were carried out in the temperature range of 5-350 K by means of MPMS-XL7 and PPMS DynaCool 9T devices. It was found that a relatively small change in the composition of the Tb-Co layer results in a sizeable change in the hysteresis loops of bilayers measured both along and perpendicular to the plane of the samples (Fig. 1). A change in the temperature of the sample is accompanied by a change in the sequence of magnetization reversal of the layers. In addition, at a certain temperature, the value of which depends on the composition of the Tb-Co layer, instead of layer-by-layer switching, magnetization reversal of the Tb-Co/FeNi film occurs as a whole structure magnetization reversal. An analysis of the obtained results suggests that in the course of deposition of the Tb-Co layer onto the FeNi layer, an interlayer magnetic boundary is formed, the parameters of which depend on the composition of the Tb-Co layer. The features of this boundary determine the value of the exchange bias and its temperature dependence. This work was financially supported by the Russian Science Foundation (RSF), project No. 18-72-10044.

**References:**

[1] C. Vogler, M. Heigl, A.-O. Mandru, et al. Phys. Rev. B 102 (2020) 014429. [2] K. Umadevi, A. Talapatra, J. Arout Chelvane, et al. J. Appl. Phys. 122 (2017) 065108. [3] V.O. Vas’kovskii, A.N. Gor’kovenko, et al. Phys. Met. Metallogr. 120 (2019) 1055. [4] E.E. Fullerton, J.S. Jiang, S.D. Bader. J. Magn. Magn. Mater. 200 (1999) 392. [5] D. Chumakov, R. Schäfer, D. Elefant, et al. Phys. Rev. B 66 (2002) 134409. [6] G.I. Frolov, V.Yu. Yakovchuk, et al. Tech. Phys. 50 (2005) 1605. [7] W.C. Cain, M.H. Kryder. J. Appl. Phys. 67 (1990) 5722. [8] K. Chen, A. Philippi-Kobs, V. Lauter, et al. Phys. Rev. Appl. 12 (2019) 024047.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7db57c54aa8f8588e53c294b6b60f8c0.jpg)
Figure 1. The hysteresis loops measured at T = 5 K in the field oriented along (a,b,c) and perpendicular (d,e,f) to film plane for Tb-Co/FeNi films with different Tb-Co layer composition.

 Spin reorientation transition and exchange bias in hard/soft Tb-Co/FeNi films

A model of exchange bias in polycrystalline bilayers based on a thermal fluctuations [1] was extended to account for an arbitrary magnetization angle of the FM layer. It allows a more accurate description of the behavior of the thermally activated reversal of the Néel vector of a grain since the extended model takes into account a variation of barrier height relative to the magnetization angle of the FM layer. Field annealing (setting of the exchange bias) and further relaxation of the FM/AFM bilayer was simulated using a sequence of iterations based on this model. At each iteration step we tracked the average value of exchange bias field amplitude and its angle, until a convergence was reached. We defined the array of grains in an AFM layer as having a lognormal distribution of grain volumes [2] and a uniform in-plane distribution of the uniaxial anisotropy axis. We have chosen the parameters of the lognormal distribution based on the fit to the experimentally-measured temperature dependence of exchange bias in our FM/AFM samples. We observed a small angle deviation from the initial field annealing direction due to the relaxation of exchange bias in a finite ensemble of AFM grains. Applying a relatively strong magnetic field at an angle to exchange bias (i.e. non-collinear depinning) for a certain time and then releasing it, gives rise to a shift of the final angle of exchange bias and a slight variation of its amplitude. Generally, the direction of the exchange bias is more affected by the FM layer non-collinear depinning than its amplitude, which recovers within a certain time interval upon release of the external field. Sweeping the angle of applied field from - to and back we observed the hysteretic behavior of exchange bias. Again, the exchange bias angle is mainly affected, rather than its amplitude. Angular variation of the exchange bias degradation and its time, field and temperature dependencies were also simulated and analyzed in the framework of this extended model.

**References:**

[1] H. Xi and R. M. White, J. Appl. Phys. 94, 5850 (2003). [2] K. O’Grady, L. E. Fernandez-Outon, and G. Vallejo-Fernandez, J. Magn. Magn. Mater. 322, 883 (2010).

**Images:**

 Angular deviation of the exchange bias in bilayer CoFe/IrMn under rotating magnetic field

High-pressure torsion (HPT) as a severe plastic deformation (SPD) technique is a well-known method of material processing [1,2]. A key feature of SPD techniques per se is the constancy of sample’s volume and shape during the deformation process and therefore allowing the production of nanocrystalline samples exhibiting bulk dimensions. One of its further advantages is the avoidance of many processing limits of binary and ternary systems that exhibit large miscibility gaps in their thermodynamic equilibrium. It allows the production of alloys with non-equilibrium compositions at low temperatures [3]. In this study, different compositions of Fe and Cr are processed with the principal goal of having a bulk material of alternating phases of ferromagnetic Fe and antiferromagnetic Cr on the nanoscale, enabling exchange bias. By HPT, the production of a supersaturated nanocrystalline FeCr alloy is possible. During a subsequent annealing treatment, decomposition might occur in closely spaced phases of separated Fe and Cr. Thus, it offers the possibility to induce exchange bias phenomena after field cooling below the Néel temperature. The nanocrystalline microstructure of the HPT-deformed samples increases the amount of interfaces; thus, increasing the effect. In a previous study of Weissitsch et al. [4] on Fe-Cr materials, it has been shown that homogeneous nanocrystalline microstructures can be produced by HPT at room temperature for compositions inside the miscibility gap. The nanocrystalline nature of the material and the identical crystal structure of Fe and Cr at room temperature (body centred cubic) with almost identical lattice parameters, make the microstructural investigation of the HPT-deformed material very challenging. Synchrotron results are ambiguous in peak deconvolution due to peak broadening and the above mentioned similarity in crystal structure. Scanning electron microscopy, including space-resolved chemical characterization, is difficult to perform due to the strongly refined microstructure. Still, the initial state has to be characterized before any further annealing treatments. An alternative characterization way is the measurement of magnetostrictive properties of the as-deformed alloys. Bormio-Nunes et al. [5] measured a large, positive magnetostriction for arc-melted supersaturated single phase FeCr alloys with up to 30 at% Cr. An increase in saturation magnetostriction with increasing Cr-content is further reported. Thus, it is of interest to study magnetostrictive properties of materials with even higher Cr-content. The saturation magnetostriction of Fe with 30% Cr (3/2λs ~ +30 to +70 ppm) found in [5], is slightly higher compared to our measurements (see Figure 1), bearing in mind the completely different involved microstructures. In the present study, the maximum saturation magnetostriction is found for 30 at% Cr as well. For an even higher Cr content of 50at%, a positive but decreasing saturation magnetostriction was measured. For a material only containing 30at%Fe, magnetostriction is about zero, which is in accordance with the very small values reported on pure Cr [6]. Due to this findings, the largest magnetostriction would have to be searched for compositions close to 30 at% Cr. Figure 1 depicts the pronounced difference magnetostrictive behaviour of FeCr compared with of pure Fe (3/2λs ~ -14 ppm). Thus, it can be stated that Fe and Cr are in a supersaturated state after HPT deformation at room temperature. To achieve an exchange bias coupled material, an improved understanding of the decomposition behaviour of the HPT-deformed FeCr alloys is crucial. Therefore, the influence of annealing treatments on the microstructural evolution and magnetostrictive properties is further investigated in this study. Annealed samples (1h at 500°C) do not show pronounced changes in magnetostrictive behaviour, i.e. the supersaturated state seems to prevail. To achieve the goal of exchange coupling, a proper concatenation of annealing treatments at even higher temperatures and subsequent HPT-deformation, producing a nanocrystalline microstructure again, is the next step. This study demonstrates that magnetostrictive measurements are an efficient tool to characterize the microstructure of nanocrystalline FeCr alloys deformed by HPT deformation, which are difficult to investigate by other techniques. Large magnetostriction is found in the as-deformed and annealed states of Fe-rich materials. HPT deformation combined with subsequent annealing treatments is a first step towards processing of Fe-Cr composites showing exchange bias phenomena. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 757333).

**References:**

[1] R. Valiev, R. Islamgaliev, I. Alexandrov, Progress in Materials Science. 45 (2000) 103–189. [2] R. Pippan, S. Scheriau, A. Taylor, M. Hafok, A. Hohenwarter, A. Bachmaier, Annual Review of Materials Research. 40 (2010) 319–343. [3] K.S. Kormout, R. Pippan, A. Bachmaier, Advanced Engineering Materials. 19 (2017). [4] L. Weissitsch, M. Stückler, S. Wurster, R. Pippan, A. Bachmaier, Materials Science Forum. 1016 (2021) 1603–1610. [5] C. Bormio-Nunes, J.P. Serra, F.S. Barbosa, M.B.S. Dias, R.S. Turtelli, M. Atif, R. Grossinger, IEEE Trans. Magn. 52 (2016) 1–4. [6] E.I. Kondorskii, T.I. Kostina, V. Yu. Galkin, Sov. Phys. JETP 42 (890) 890-891.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/c218e9438d71e8e2fddeb74c82210c55.jpg)
Figure 1: Magnetostriction as a function of applied field, measured for HPT-deformed FeCr materials, spanning the composition range from 30at% to 70at% Fe. A comparison with the markedly different magnetostrictive behaviour of pure Fe is given.

 Severe plastically deformed, supersaturated FeCr alloys – a candidate material for exchange bias and enhanced magnetostriction

We study the extrinsic factors influence on FMR LW in polycrystalline systems F/AF with exchange bias, by varying AF layer thickness for different sequence of F and AF layers deposition. We observe that the exchange bias has a negligible effect on FMR LW. Instead, the contributions of uniaxial anisotropy, microstructure and F/AF interface roughness can be of major extrinsic factors. Using DC magnetron sputtering, Ta(30nm)/NiFe(10nm)/IrMn(tAF)/Ta(30nm) (TS) and Ta(30nm)/IrMn(tAF)/NiFe(10nm)/Ta(30nm) (BS) multilayers on silicon substrate were prepared with the AF thickness tAF = 10, 20, 30 and 40 nm. A permanent in-plane magnetic field of 420 Oe was applied during the deposition. Sample surface morphology was investigated by atomic force microscopy to obtain the mean square roughness σrms. The magnetic characteristics were investigated using FMR at a constant frequency ω/2π = 9, 65 GHz and with magnetic field H directed along the sample surface and scanned in the range up to 6 kOe. The main magnetic characteristics of the samples were obtained from the angular dependences (AD) of the resonant magnetic field, Hr(α),where α is the angle between the direction of the FMR field and the direction of the magnetic field applied during the deposition of the sample. Despite some quantitative differences, the qualitative form of the dependency of the exchange bias on the AF layer thickness is the same for both TS and BS structures. However, there is a significant discrepancy in the widths of the FMR lines for TS and BS structures: ΔHTS is smaller than ΔHBS. In addition, in the case of TS, the values of ΔH for α = 0o and α = 90o coincide within the experimental accuracy and synchronously repeat the variations with thickness. These observations indicate that the effect of the exchange bias on the FMR LW is negligible small compared to the influence of other factors. The roughness, σrms, for TS samples is greater than for BS, the dependence σrms,(tAF) is nonmonotonous and is in antiphase for TS and BS structures. The effect of the sample surface roughness on the FMR LW is quite noticeable. The experimental data can be approximated for TS samples by the dependence ΔHTS ~ (σrms S)1/3 , where S is the average area of defects and the FMR line width is propotional to the average linear size of defects on TS sample surface. This tendency is qualitatively consistent with the known results [1], where a linear relationship between ΔH and the pore radius on the sample surface was predicted. In BS samples, as compared with TS, the picture is quite different. The roughness is weaker while the LW is larger. That assumes that the roughness does not play a dominant role in LW of BS samples. Evidently, the features of the layers growth lead to different magnetic properties of AF/F and F/AF samples at the same chemical content of the F and AF layers. When the F layer grows on the AF layer, the microstructure of the AF layer can affect the properties of the F layer deposited on it. An increase in the AF layer thickness can lead, for example, to an increase in the size of AF grains in the bulk and at the AF/F interface. The F layer deposited on top takes on a typical AF grain size, i.e., the F grain size, possibly, increases with increasing AF layer thickness. In this case, the uniaxial magnetocrystalline anisotropy HK becomes the dominant factor affecting the broadening ΔH. Taking into account that the samples have a polycrystalline structure, we assume that the approximation of strongly bonded grains with HK &lt;&lt; 4πMs works. In this approximation, a polycrystal is regarded as a homogeneous medium with an inhomogeneous effective field, which is affected by the presence of grains with different crystallographic orientations. The magnetic fluctuations of the grains are coupled to each other through demagnetizing fields [2], which leads to uncompensated crystallographic anisotropy, which enhances uniaxial anisotropy and causes smooth spatial oscillations of magnetization. Spatially fluctuating magnetization, the so-called “micromagnetic ripple” observed in Lorentz transmission microscopy [3, 4], correlates with the grain size [3] and can be, if not the main, then an important additional source of broadening of the FMR line [5]. We show that FMR LWs in the BS samples decrease with increase of AF layer thickness, approximately, as 1/tAF β, with β = 2±1 and the difference in LW values measured along HA and EA in BS samples varies proportionally to the square of HK.

**References:**

1. M. Sparks, R. Loudon, and C. Kittel, Phys.Rev. vol. 122, p.791 (1961) 2. A.G. Gurevich, G.A. Melkov. Magnetization Oscillation and Waves, CRC Press, New York (1996). 3. N.G. Chechenin, C.B. Craus, A. Chezan, et al IEEE Trans. Magn. MAG 38, p. 3027 (2002). 4. N.G. Chechenin. J. Magn. Magn. Mater. Vol. 300, 198 (2006). 5. N.G. Chechenin, Phys. Solid State vol. 46, p. 479 (2004)

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