Planar array of self-assembled Ga$_{x}$Fe$_{4-x}$N nanocrystals in GaN: Magnetic anisotropy determined via ferromagnetic resonance

The magnetic anisotropy of a planar array of Ga$_{x}$Fe$_{4-x}$N nanocrystals (NCs) embedded in a GaN host is studied by ferromagnetic resonance. X-ray diffraction and transmission electron microscopy are employed to determine the phase and distribution of the nanocrystals. The magnetic anisotropy is found to be primarily uniaxial with the hard axis normal to the NCs plane and to have a comparably weak in-plane hexagonal symmetry. The origin of the magnetic anisotropy is discussed taking into consideration the morphology of the nanocrystals, the epitaxial relations, strain effects and magnetic coupling between the NCs.


INTRODUCTION
(TMGa) and ferrocene (Fe(C 5 H 5 ) 2 ), the flow rates during the deposition of the buffer and of the capping layer are 1500 standard cubic centimeters per minute (sccm) for NH 3 , and 25 sccm for TMGa. In the Fe-doped layer the NH 3 flow rate is reduced to 800 sccm, while the ferrocene flow rate is 490 sccm. When the TMGa source is enabled for the digital doping, it provides a flow rate of 5 sccm.
Transmission electron microscopy (TEM) techniques have been employed to establish the crystallographic phase, orientation and distribution of the nanocrystals in the GaN matrix [9]. Cross-section and plan-view TEM specimens are prepared by mechanical polishing, The magnetic resonance measurements have been performed with a Bruker Elexsys E580 electron paramagnetic resonance spectrometer capable of static magnetic fields up to 1.5 T and equipped with a continuous flow cryostat. An X-band microwave cavity is employed and the measurements are carried out at microwave frequencies between 9.4 and 9.5 GHz. The static magnetic field has been modulated with an amplitude of 0.5 mT at 100 kHz to allow lock-in detection. The samples are cut into 4 mm 2 square specimens necessary to fit the sample space in the spectrometer for in-plane experiments. The out-of-plane measurements have been carried out by recording an FMR spectrum every 10 • for one full rotation. Inplane measurements have been performed every 2 • for half a circle. The microwave power has been adjusted to 2 mW for the in-plane measurements, while due to larger peak width 20 mW have been employed for the out-of-plane experiments.

EXPERIMENTAL RESULTS
In the TEM micrograph of a cross-section specimen reported in Fig. 1a, it is shown that the NCs distribute on a plane perpendicular to the c growth direction, over a thickness of ∼40 nm. Moreover, as estimated from a statistically significant number of plan-view images similar to Fig. 1b, the average diameter of the NCs is (24.7±5.2) nm. A similar analysis of 14 cross section TEM images yields an average size of (16.9 ± 2.2) nm along the c-direction of GaN and of (20.1 ±3.9) nm perpendicular to it. The nanocrystals are estimated to occupy    Temperature dependent FMR measurements have been carried out, and the signal clearly visible at room temperature (RT) broadens with decreasing temperatures, quenching around 40 K. Therefore, the anisotropy measurements reported here are restricted to RT. In this context, effects of thermal broadening may be considered [17] and a similar temperaturedependent behaviour was reported by Bardeleben et al. for Co precipitates in ZnO and attributed to the nanocrystalline structure of the films [18].
Due to the equivalent in-plane orientations of the nanocrystals analysed here, one would expect to detect three FMR lines, which coincide when the magnetic field is perpendicular to the sample surface (θ = 0). Nevertheless, only one line can be observed, and shows an uniaxial dependence (cos 2 (θ)) when varying the out-of-plane angle θ, while measurements with the magnetic field in-plane (θ = π/2) evidence a 6-fold symmetry with an angular dependence close to sin (6φ). When the field is nearly perpendicular to the sample surface,

Resonance field [T]
In plane angle ϕ [rad] the absorption peak is much broader than for angles with the field nearly in-plane. The angular dependence of the resonance field is reported in Fig. 3.

DISCUSSION
In order to shed light on the origin of this uniaxial angular dependence, we consider the shape of the nanocrystals as determined by HRTEM, since any non-symmetric uniformly magnetized particle will show uniaxial shape anisotropy. By approximating the nanocrystals to oblate spheroids [19] and using the nanocrystals dimensions mentioned above, the anisotropy tensor in diagonal form has the following components: N xx = N yy = 0.31 ± 0.03 and N zz = 0.38 ± 0.06. The error bars of the shape anisotropy do not rule out that in average the NC shape could be isotropic. In this case, strain induced by the GaN host crystal on the cubic Ga x Fe 4−x N NCs can serve as an explanation for the observed uniaxial anisotropy, since it would generate uniaxial terms in the free energy, which, assuming a high enough prefactor, could dominate over the cubic terms from the iron nitride crystal lattice.
Alternatively, a sizable magnetic coupling between the nanocrystals supported by the planar arrangement would as well lead to uniaxial anisotropy [18]. ) and Fe 4−x N (whose value was experimentally established to be 1.42 × 10 6 A/m in thin films [22], and 1.51 × 10 6 A/m in powders [3,23,24]). Based on these figures, the coverage of (4.8±0.2)% obtained by (HR)TEM for the samples considered in this work, represents a border limit for the observation of coupling between the NCs at room temperature.
In order to elucidate the origin of the observed in-plane magnetic anisotropy it is mandatory to take into consideration a number of factors. Shape anisotropy can be ruled out, since for single domain particles only uniaxial like shape anisotropies are enabled. While the phenomenologic theory of magnetic crystal or strain anisotropy does not exclude the appearance of high order terms, which would not cancel out for the given strain geometry, microscopic arguments speak against strain as possible explanation, since if a quadratic lattice is isotropically stressed along three directions -with an angle offset of 120 • between them -the resulting lattice will still be a square one. Also, due to the small size of the nanocrystals considered, relaxation of strain is not to be expected, therefore a local change in magnetic anisotropy within a NC is unlikely. On the other hand the observed 6-fold anisotropy is compatible with the arrangement of a minority of the nanocrystals with their in the Additional Material. In order to determine the resonance frequency, the method described by Smit and Beljers [26] is employed, which allows to calculate the frequency ω for FMR conditions from the free energy F , written as a sum over the different anisotropy contributions and with γ as gyromagnetic ratio: In the fit, the equilibrium magnetization direcion (θ, φ) has been determined by minimizing the free energy for each computation of the resonance frequency, while the damping α has been neglected. All five models describe the observed data well, the best agreement is nevertheless obtained with the purely phenomenologic hexagonal crystal anisotropy model. The g-factor, the saturation magnetization and the anisotropy constants of the crystal anisotropy have been fitted and the values are reported in Table I. Since the uniaxial anisotropy generated by the NC shape, the saturation magnetisation and the uniaxial crystal field anisotropy term are mutually dependent, the obtained quality of the fit is commensurate for the models including the phenomenologic hexagonal term but without the fourth order uniaxial term, and for the two models assuming the cubic anisotropy. The variation is restricted to the fitted anisotropy constant values.
While the obtained g-factor differs from the value reported for Fe 4 N thin films [27], in the frame of the hexagonal model which is not including the shape anisotropy the fitted saturation magnetization of 1.44 × 10 6 A/m is very close to the literature values 1.42 ×  I: Material parameters obtained by fitting the angular dependence of the FMR (details are provided in the Additional Material). In the models (i), (ii) and (iii) the nanocrystals are treated as isolated magnetic moments. The model (i) is purely phenomenologic and includes a uniaxial/hexagonal anisotropy, where K 1 and K 2 are the prefactors of the second and fourth order uniaxial term, and K 3 is the prefactor of the sixth order hexagonal term; in (ii) the pre-factors of the uniaxial crystal strain are set to zero, while the model includes the shape information from HRTEM for shape anisotropy; for (iii) the shape information from HRTEM is employed and the nanocrystals are modeled with a cubic anisotropy and with their [111] direction parallel to the [001] direction of GaN, with K 1 and K 2 being the cubic anisotropy term prefactors of second and fourth order. For the rigidly coupled models the shape anisotropy is set to the one of an infinitely extended thin layer.

Individual Coupled
Crystal/Strain Shape + Hex Shape + Cube Hexagonal Cubic incorporation of Ga into the NCs, which would lead to a decreased saturation magnetization, cannot be ruled out. Regarding the models assuming rigidly coupled nanocrystals, the saturation magnetization is much weaker than the one reported for thin films. This may be attributed to the fact that in the assumed arrangement of coupled NCs a substantial fraction of the volume is actually represented by paramagnetic dilute (Ga,Fe)N, as proven by SQUID magnetometry measurements [28]. By calculating the ratio between the literature value and the measured saturation magnetization and neglecting the paramagnetic contribution, approximately 3% of the film volume is found to consist of Ga x Fe 4−x N, comparable to the value obtained from (HR)TEM.
As stated above, shape anisotropy can fully explain the out-of-plane anisotropy, yet the average dimensions yield a relatively low saturation magnetization. Since the model neglecting the shape anisotropy and describing the system with crystal/strain provides a significant agreement between the fitted saturation magnetization and the literature value, an estimation of the strain that would be required to obtain the observed uniaxial anisotropy of K 1 = (−40492 ± 495) J/m 3 has been carried out. According to Ref. [21], the relation between strain and anisotropy is given by: where λ S is the saturation magnetostriction, E the Young modulus and ǫ the strain. can also be described within a conventional hexagonal magnetic crystal anisotropy model, and a significant agreement is obtained for a layer of coupled crystals.
A broad range of applications for metallic and magnetic nanocrystals in a semiconductor matrix is envisaged and among the most thrilling prospects one can mention the option of exploiting them as spin current injectors into the semiconductor host crystal [13], possibly via FMR spin pumping [14,15], and spin current detection by inverse spin Hall effect [35]. A further application is directed to electric flash memory-like data storage [36,37]. For magnetic storage in-plane anisotropy is required, and it can be expected to be induced by e.g. elongated nanocrystals obtained by modulating the growth conditions [38]. Moreover, the control of the magnetic coupling between nanocrystals makes the hybrid semiconductor/NCs system an exciting candidate to study frustrated magnetic systems and spin glass behaviours [20]. The FMR frequency is derived from the models using the formula of Smit and Beljers [1], evaluated at the equilibrium magnetization direction. The free energy used in the models (i)-(v) fitted to the data reads as follows: where F Z denotes the Zeeman energy, F S the shape anisotropy energy, and F C the crystalline or strain anisotropy. Additional terms, like surface-and step-anisotropy are not taken into account.
The three contributions are given by: Here µ 0 is the Bohr Magneton, M is the magnetization and H is the applied external field.
The tensor N is the shape anisotropy tensor, rotated to match the coordinate system of the crystal anisotropy.
The formula to be employed for crystal anisotropy depends on the symmetry of the crystal lattice. Here, cubic and hexagonal anisotropies are considered. These anisotropies are generally expressed in a phenomenological model which is built using a Taylor expansion of the free energy with respect to the direction cosines a i . For cubic and hexagonal crystals Strain anisotropy is formally equivalent to uniaxial shape anisotropy (first term of the hexagonal anisotropy) and will not be covered separately, considering that the link to strain has already be mentioned in the main text.
The models used in the text are: • (i) hexagonal crystal anisotropy: • (ii) hexagonal crystal anisotropy + Shape information from HRTEM: The shape anisotropy tensor is calculated from the size information obtained by HRTEM, using the formulas for an oblate spheroid [2].
• (iii) cubic crystals with their [111] direction along [001] of the GaN host: the same shape anisotropy tensor as above, rotated with respect to the crystal axis with Euler angles −π/4, and − arcsin 2/3 , while the third Euler angle is the in-plane rotation of the crystals, which is adjusted to get the best agreement between measurement and model.
• For the rigidly coupled models (iv) and (v) the shape anisotropy is replaced by the shape anisotropy tensor for an infinitly extended thin film, having only one component N zz . The formalas from above are employed.