The SR-XRD 2θ-θ scans confirm the formation of Mn₅Ge₃ nanomagnets. As shown in Figure 1, beside the main peaks from Ge(004), the diffraction peaks of Mn₅Ge₃(111), (002), (310), (222), and (004) are clearly visible. Note that, compared to the work by Ottaviano et al. 20, the SR-XRD reveals more Mn₅Ge₃ peaks even for a much smaller Mn ion fluence due to the large flux of X-rays from the synchrotron source, which allows for the detection of small Mn₅Ge₃ nanocrystals. Therefore, we have to revisit the work by Ottaviano et al. They concluded that Mn₅Ge₃ nanocrystals formed by ion implantation are preferentially (002)-oriented in the Ge(001) matrix 20. However, Zeng et al. 10 prepared Mn₅Ge₃ layers by molecular beam epitaxy, and they found the crystalline orientation as Mn₅Ge₃(001) ∥ Ge(111). The SR-XRD observations therefore led us to conclude that Mn₅Ge₃ nanocrystals formed by Mn implantation are indeed randomly oriented inside the Ge(001) matrix, which is also supported by the magnetic properties shown later. We found nearly isotropic hysteresis loops with magnetic field along different directions. Note that in the work of Jain et al. 26, the Mn₅Ge₃ nanocrystals were grown by annealing GeMn films on Ge(001) substrates prepared by MBE. Most of the Mn₅Ge₃ clusters (97%) have their c-axis perpendicular to the film plane. The accumulated literature data suggest that the growth of Mn₅Ge₃ nanocrystals from the Ge matrix is different from the Mn₅Ge₃ thin films.

Figure 2 shows cross-section TEM images of samples 1E16 and 5E16. The white contrast spots are from precipitates that are located in the depth between 20 and 120 nm, which is in agreement with the depth profile of Mn ion implantation. The average crystallite size is increased from 5 to 11 nm with increasing Mn fluences from 1 × 10¹⁶ to 5 × 10¹⁶ cm⁻². For detailed analysis, we focus on the sample 5E16. As shown in Figure 2b, the well-defined Moiré patterns are a strong indication for monocrystalline precipitates embedded in a crystalline matrix. Using high-resolution TEM, the precipitates can be identified to be Mn₅Ge₃, as shown in Figure 2c,d. Figure 2d is the fast Fourier transform (FFT) of the image shown in Figure 2c. The FFT reveals lattice spacings amounting to 0.298 nm (indicated by the open circles) and 0.623 nm (indicated by the open squares), which correspond to Mn₅Ge₃(111) and (001), respectively.

Figure 3a shows the zero field-cooled and field-cooled (ZFC/FC) magnetization curves in a 50-Oe field for different Mn fluences. The FC curve for sample 1E15 completely overlaps with the corresponding ZFC curve at around zero. Magnetic Mn₅Ge₃ nanocrystals can be excluded in this sample, which is consistent with the SR-XRD observation, except that they are very small and dilute beyond the detection limit of SR-XRD. For samples 1E16 and 5E16, a distinct difference in the ZFC/FC curves was observed. The ZFC curves show a gradual increase at low temperatures, peaking at different temperatures, while the FC curves monotonically increase with decreasing temperature. The width of the peaks in the ZFC curves is due to the size distribution of Mn₅Ge₃ nanocrystals, as shown in the TEM images (Figure 2). In this paper, we take the temperature (T_max) at the maximum of the ZFC curve as the average blocking temperature listed in Table 1.

Figure 3b shows the magnetization versus field reversal (M-H) of all samples measured at 5 K. Hysteretic behaviors were observed for samples 1E16 and 5E16. With increasing Mn concentration, the saturation magnetization is increased from 10.1 to 69.2 emu/cm³ (by assuming the implanted depth of 100 nm), and the coercivity is increased from 0.22 to 0.26 T. At 300 K, sample 5E16 only shows field-induced magnetization (see the inset of Figure 3b). The saturation magnetization of the sample 5E16 (1E16) is 69.2 (10.1) emu/cm³, corresponding to around 1.5 (1.1) μ_B/Mn, which is smaller than 2.6 ± 0.5 μ_B/Mn as reported in the study of Bihler et al. 1, which means that not all of the implanted Mn ions form the ferromagnetic Mn₅Ge₃ phase.

We also compared the magnetization between the in-plane and out-of-plane directions at 5 K for sample 5E16 (not shown). In contrast to the studies of Bihler et al. 1 and Jain et al. 8, there is no detectable magnetic anisotropy. For the bulk Mn₅Ge₃, the magnetic easy axis is [001]. The absence of magnetic anisotropy in our samples is due to the random crystallographic orientation of the Mn₅Ge₃ nanocrystals.

As shown in Figure 3b, the hysteresis loop is not square-like. The distribution of coercivity field is due to the size distribution of the nanomagnets, as evidenced by the TEM images, and is also possibly due to the random distribution of the nanomagnet easy axis. According to the Stoner and Wohlfarth model for single-domain magnetic nanoparticles, the maximum coercive field gives the anisotropy field μ₀H_a2 = 0.26 T for sample 5E16. Using the bulk saturation magnetization (M_S) for Mn₅Ge₃ (1,100 kA/m) 26, one can deduce the anisotropy constant: K₂ = μ₀H_a2M_S/2 ≈ 1.4 × 10⁵ J/m³, which is smaller than the value reported by Jain et al. 26. Based on the Néel-Brown model, the volume for a nanomagnet V = 25k_BT_max/K₂ (k_B as the Boltzmann constant), we calculate the average diameter of Ge₃Mn₅ clusters in sample 5E16 to be approximately 10.8 nm (T_max = 270 K). The average diameter is in good agreement with the results obtained by TEM. However, the average diameter for sample 1E16 deduced from the ZFC magnetization is as large as 9.5 nm, which is much larger than the value from the TEM observation.

The magnetic properties of the Mn-implanted Ge were also investigated by X-ray magnetic circular dichroism (XMCD) at Mn L_2,3 edge. Right before the XAS measurements, the sample was etched in deionized water for 2 min to remove the surface oxide layer 27. Figure 4a presents the Mn L_2,3 XAS measured in total electron yield mode at around 4.5 K; μ + and μ − represent the absorption intensity with the direction of magnetization parallel and antiparallel to the photon helicity, respectively. As shown in Figure 4a, after etching, we obtained very similar spectra as was reported for ferromagnetic Mn₅Ge₃ 28. The XAS spectra can be classified into the 2p_3/2 (approximately 641 eV) and 2p_1/2 (approximately 651 eV) absorption regions. The shape of the main feature indicates the itinerant nature of ferromagnetic Mn₅Ge₃. On the other hand, the weak shoulders appear at 642 and 644 eV, and the doublet structure is observed in the 2p_1/2 excitation region, which could be related with some diluted Mn impurities in the Ge matrix 21 or oxidized Mn 29. Figure 4b shows the XMCD spectrum, revealing a large negative signal (approximately 641 eV) and a small positive signal (approximately 644 eV) in the 2p_3/2 region and a larger positive signal (approximately 651.5 eV) in the 2p_1/2 region. Note that the shoulders and the doublet in XAS spectra are hardly resolvable in the XMCD spectrum, which indicates that the oxidized Mn ions have no contribution to the ferromagnetism. According to the sum rule, the integrated intensity of the XMCD signal in the whole region is proportional to the orbital magnetic moment relative to the spin magnetic moment. In the present XMCD spectrum, the integration is nearly zero, indicating that the orbital magnetic moment is negligible for Mn₅Ge₃. Comparing our experimental results with the published calculations, the electronic structure of Mn ions can be assumed to be in the 3d⁶ configuration without spin-orbit interaction 2830.

All three samples show p-type conductivity and large magnetoresistance (MR) effect. Figure 5a shows the measurement results for sample 5E16. The MR is defined as R(H) − R(0)] / R(0), where R(H) is the sheet resistance at a field of H, R(0) is the sheet resistance at zero field. One can see that MR is positive and may saturate at a large field. Different from ferromagnetic semiconductors or metals, there is no hysteresis in MR curves for the Mn₅Ge₃/Ge hybrids. In this case, the spin scattering should have a very small contribution to transport. Similar positive MR effect has also been reported for GeMn-nanocolumns/Ge hybrids 15. Note that the Ga-doped Ge shows only the neglectable MR effect as shown in Figure 5a. The MR effect can be interpreted by the inhomogeneity of the sample: the different conductivity and Hall resistivity of GeMn and Ge (or Mn-rich and -poor regions). We modeled the hybrid system where Mn-rich nanoparticles were embedded in the Ge matrix by a 2D slice as in the work of Yu et al. 31. Under steady-state conditions, the continuality of the current requires that ∇ · σ · ∇ U x , y = 0 , where U x , y is the electrostatic potential at position (x, y) on the 2D slice. The materials were fully characterized by their conduction matrices, which vary with the position of different materials. Thus, the finite element method (FEM) proposed by Moussa et al. 32 was used. We applied a constant potential between two electrodes and calculated the induced averaged potential difference at the other two electrodes in the geometry of the van der Pauw method. The current normal to the boundary of the slice was set to zero (the natural boundary condition). The transport properties of the matrix and nanocrystal are simple, characterized by the conductivity matrix with the components:

σ xx β = σ yy β = σ 0 1 + β 2 ,

σ xy β = − σ yx β = − σ 0 β 1 + β 2 ,

where σ(0) is the zero-field conductivity and β = R_Hσ(0)μ₀H, in which R_H is the Hall coefficient of the materials and μ₀ is the susceptibility in vacuum.

The material parameters of the matrix are chosen to be σ H Ge 0 = 10 4 Ω m − 1 , R H Ge = 10 − 6 C − 1 m 3 . The material parameters of the nanocrystal are as follows: σ H GeMn 0 = a σ H Ge 0 and R H GeMn = b R H Ge . There are two free parameters a and b which are the ratios of conductivity and Hall coefficient of the two phases, respectively. Both the conductivity and the Hall coefficient are functions of temperature. The resistance of the system is calculated by FEM where a constant current is applied and the corresponding voltages are measured in the geometry of the van der Pauw method. The calculated curves are presented in Figure 5b. The experimental MR curves can be well reproduced by FEM calculations. The a and b values used in the FEM calculations are shown in Figure 5c. The MR magnitude is sensitive to the ratio of conductivity of the two constitutes. Beside the magnetoresistance, the samples also show anomalous Hall resistance (i.e., the Hall resistance deviates from a linear behavior), which can be explained by two kinds of carriers with different mobilities 33. On the other hand, we have to note the rather large discrepancy in the MR magnitude between the experimental and modeled values. In the model, for simplifying, we neglect the anomalous Hall effect in the GeMn constitute, which may induce this discrepancy. Also, in order to account non-monotonic dependence of MR on temperature (see Figure 5c), we have to vary parameters a and b accordingly. The decrease of a and b at temperature below 50 K cannot be understood and is the aim for the future work.