The high-resolution transmission electron microscopy (TEM) and scanning TEM (STEM) experiments were performed on a FEI Tecnai F20 (S)TEM operating at 200 kV. The digital images were recorded by a Gatan^® 2k × 2k CCD camera. All the TEM and STEM images were taken in standard conditions. However, it should be noted that the MnGe nanodots appear dark contrast in the bright-field TEM mode (Figure 2a, c, d) which is different from the case in the low-angle dark-field STEM mode (Figure 2b, e) where the MnGe show white contrast due to different imaging systems.

A physical property measurement system and superconducting quantum interference device was used to measure the magnetotransport and magnetic properties, respectively. Both equipments were manufactured from Quantum Design.

Practically, we employed a concept of stacked MnGe nanodots by alternatively growing MnGe and Ge layers with designated thicknesses (nominal 3-nm-thick MnGe and 11-nm-thick Ge), as shown in Figure 1a. It is well known that Mn doping in Ge induces compressive strain because of its larger atomic size [²⁹], assuming that no lattice defects are generated during the doping process, i.e., lattice coherence. Mn-rich MnGe nanodots induced by the spinodal decomposition should be strained if the lattice coherence between the nanodots and the matrix remains. Once the strained nanodots are developed, a thin Ge spacer layer, subsequently deposited with an optimized thickness, will retain the perfect lattice coherence with the underneath nanodots. This enables the existing nanodots to exert strain on the Ge spacer layer and produce "strained spots", which, in turn, become preferred nucleation sites for successive nanodots. Eventually, multilayered and vertically aligned MnGe nanodot arrays can be produced, similar to the scenarios of stacked InAs/GaAs [²⁷,²⁸] and Ge/Si [³⁰] quantum dots. Indeed, by employing this innovative approach, we achieved the growth of coherent self-assembled MnGe nanodot arrays with an estimated density of 10¹¹ cm^-2 to approximately 10¹² cm^-2 within each MnGe layer, as schematically demonstrated in Figure 1b. In this study, ten periods of MnGe nanodots were epitaxially grown on Ge (100) and GaAs (100) substrates by a Perkin-Elmer solid source molecular beam epitaxy (MBE) system. A detailed description of growth method and parameters are presented in the Methods part (also refer to Figure S1 in Additional file 1). TEM and energy dispersive spectroscopy (EDS) in the STEM mode were performed to understand the nanostructures and compositional variations of the resulting thin films. Figure 2a and 2c are typical plane-view and cross-sectional TEM images and show the general morphology of the MnGe nanostructures, viewed along the <100> and <011> directions, respectively. A high-density of dark nanodots can be clearly seen in both cases. Based on the magnified cross-sectional image shown in Figure 2d, the nanodot arrays are clearly observed with ten stacks along the growth direction although not perfectly vertical (see Figure S2 in Additional file 1 for more images). In order to determine the composition of the dark dots, EDS analyses in the STEM mode were carried out and typical plane-view and cross-sectional STEM images are shown in Figure 2b and 2e, respectively. Figure 2f is the EDS result taken from a typical dot and shows clearly the Mn and Ge peaks. Figure 2g and 2h present EDS line scans using the Mn K peak for the dots marked by G and H in Figure 2b and 2e, respectively, indicating high concentrations of Mn inside the dots. Taking all these comprehensive TEM results into account, it is concluded that the nanodots are Mn-rich when compared with the surrounding matrix. Figure 3a shows a high magnification TEM image taken from a thin area, where several aligned MnGe nanodots can be evidently observed. The distance between two vertically adjacent nanodots (along the growth direction) is measured to be 14 ± 1 nm, well matched with the designed period of a 11-nm-thick Ge spacer layer and a 3-nm-thick MnGe layer. It should be noted that these nanodots are uniform in size with an elliptical shape (a dimension of 5.5 ± 0.5 nm and 8 ± 0.3 nm in the horizontal and vertical directions, respectively), as demonstrated in Figure 3a. Since the nominal thickness of the MnGe layer (3 nm) is far less than the dot vertical dimension (8 nm), it suggests that, during the growth of the MnGe thin film, Mn not only diffuses laterally (to form dots), but also migrates vertically into the adjacent Ge spacer layers, primarily in the proximity of the dot regions, resulting in ellipse-shaped nanodots.

To determine structural characteristics of the MnGe nanodots at the atomic level, high-resolution TEM (HRTEM) was used and an example is shown in Figure 3b, where the HRTEM image was taken from the dashed rectangle area in Figure 3a. Interestingly, a careful examination of the HRTEM image shows that the MnGe nanodots have an identical single-crystalline structure to the Ge matrix (the diamond structure) with no observed lattice defects, consistent with other reports (with irregular shape of MnGe clusters) [¹⁸,¹⁹]. As mentioned above, this type of MnGe nanodots is lattice coherent. This is substantially different from other Mn-rich precipitates such as hexagonal Mn₅Ge₃ [³¹] and Mn₁₁Ge₈ [³²] which have a different phase, other than a diamond structure as Ge matrix. This is also verified by our selected area diffraction patterns (refer to Figure S2 in Additional file 1), where no extra diffraction spots or diffused ring(s) can be observed. To further determine the possible lattice distortion of the MnGe nanodots with respect to the Ge matrix, the inversed Fourier transform (Bragg filtering) technique [¹⁹] was used where two sets of nano atomic planes are shown in Figure 3c and 3d. As can be observed, the interfaces between the MnGe nanodots (the dark areas) and the Ge matrix are perfectly coherent without noticeable lattice distortion or bending of the atomic planes. In fact, using the (111) atomic spacings away the Ge matrix as a reference, the MnGe spacing of (111) atomic planes are determined to be identical to that of the Ge matrix. A quantitative EDS analysis suggests that the dots have a Mn concentration as high as 11% (Figure 2f), which can be further adjusted by altering the Mn flux during the growth. The high Mn doping is comparable to the reported Mn concentration of 15% in Ref. [¹⁸]. Since the atomic radius of Mn (140 pm) is larger than that of Ge (125 pm) [²⁹], it is expected that these Mn-rich dots experience a compressive stress caused by the surrounding Mn-poor Ge matrix. In fact, such a stored stress can be visualized from the strong contrast of the nanodots shown in Figure 3. Therefore, the successful vertical alignment of stacked nanodots can be attributed to the strain fields induced by the underlying Ge spacer layers, which is consistent with the growth mechanism of stacked quantum dot systems.

Since the nanodot array samples are ferromagnetic below 300 K (Figure S3 in Additional file 1), it is of great interest to study their magnetotransport properties. To do this, the samples were then fabricated into standard Hall bars with a typical channel width of 500 μm. For all measurements, the external magnetic field (H) was applied perpendicular to the sample surface. In order to completely avoid the substrate (Ge) conducting effect (Figure S4 in Additional file 1) [³³], we have also successfully grown the same nanostructures on GaAs substrates under the same growth conditions as GaAs has the almost identical lattice parameter as Ge.

The resistivity measurements were carried out to probe the carrier transport under different temperatures. It was found that the temperature-dependent resistivities rapidly increase with decreasing temperature due to the carrier freeze-out effect at low temperatures, which is typically observed in doped semiconductors [³⁴]. Considering the embedded MnGe nanodots, the rise in resistivities at low temperatures also suggests a strong localization of carriers, which takes place at the Mn sites and/or at the MnGe/Ge interfaces, similar to the scenario of MnSb clusters in InMnSb crystals [⁸]. The temperature-dependent resistivity can be generally described by [³⁵]

where ρ(T) is the temperature-dependent resistivity; ρ₀ and T₀ denote material parameters, α is a dimensionality parameter: α = 2 for one-dimensional (1D), α = 3 for 2D, and α = 4 for 3D systems. In order to reveal the carrier transport mechanisms at different temperature regions, fittings were performed in the plots of lnρ as a function of T^-α (Figure 4a). The best fittings were found when α equals to 1 and 4 in the high-temperature and low-temperature regions, respectively, corresponding to the carrier transport via the band conduction [³⁶] (thermal activation of acceptors) and the 3D Mott's variable range hopping processes [³⁵]. According to the fitting results to Equation 1, the obtained nanodot arrays show a dominated hopping process below 10 K. At such a low temperature, the majority of free holes are recaptured by the acceptors. As a result, the free-hole band conduction becomes less important and hole hopping directly between acceptors in the impurity band contributes mostly to the conductivity [³⁶]. Above 100 K, the conduction is dominated by the thermal activation of the holes (the band conduction). A thermal activation energy (E_a) of 15 meV can be obtained from Equation 1 with α = 1 and E_a = T₀K_B, where K_B is the Boltzmann constant. This activation energy does not correspond to any known acceptor energy levels due to Mn doping in Ge, consistent with results shown in reference [²⁰].

To explore practical applications for our extraordinary nanodot arrays, the MR measurements were performed from 2 to 300 K with an external magnetic field up to 10 Tesla. Figure 4b shows the plots of temperature-dependent MR at given magnetic fields (5 and 10 Tesla) for the nanodot arrays. Under a strong magnetic field, the MR in the region of variable range-hopping conduction can be described by [³⁷,³⁸]

where the magnetic length λ equals to (cħ/eH)^1/2 and C is a field and temperature independent constant. Note that the Equation 2 is only valid in a strong-field limit [^37-39]. The inset in Figure 4b shows the best fitting results, in which a linear behavior of MR versus T^-1/3 is obtained, further confirming the hopping conduction mechanisms (T ≤ 8 K). Note that the absolute values of MRs were used for the fitting purpose. These fitting results are reasonably close to the obtained hopping regions determined from the zero-magnetic-field resistivity measurements (T ≤ 10 K, Figure 4a).

It is striking to observe that the coherent MnGe nanodot arrays present a large and positive MR up to 900% at 2 K (Figure 4c). Traditionally, the positive MR is attributed to the Lorentz force in the semiconductor matrix, which deflects the carriers during the transport process [³⁹]. The resulting MR is positive and proportional to (μH)² under low magnetic fields [¹⁹] (H ≤ 1 Tesla in our case) where μ is the semiconductor mobility (units m²V^-1S^-1 or T^-1) and H is the magnetic field. However, with a simple calculation, the estimated orbital MR is too small to explain the large MR observed from the nanodot arrays. Instead, we anticipate that besides the effect of orbital MR, the high-density magnetic nanodots could significantly contribute to the large MR ratios due to an enhanced geometric MR effect, from which the current path may be significantly deflected when external magnetic fields were applied to the magnetic nanostructures [¹⁹,⁴⁰,⁴¹]. To elucidate the underlying physics of the geometrical effect, we consider a thin Hall bar geometry with a measurement current applied in the x-direction, a Hall voltage in the y direction, z direction normal to the sample surface, and an external magnetic field H parallel to z. For semiconductors, the current density and the total electric field can be described by j=σ¯E, where the magneto-conductivity tensor is given by [⁴⁰,⁴¹]

Here, β = μH. At zero magnetic field, β vanishes. The conductivity tensor is diagonal when lacking of the magnetic field; and the current density can be simply described by j = σE. Since the electric field is normal to the surface of a metallic inclusion and j || σE , the current flowing through the material is concentrated into the metallic region which behaves like a "short circuit" (Figure S5 in Additional file 1) [⁴²]. As a result, the inclusion of metallic clusters can lead to a higher conduction than that of a homogeneous semiconductor [¹⁹,⁴⁰,⁴¹]. However, at high magnetic fields (β>>1), the off-diagonal terms of σ¯(H) dominate. Equivalently, the Hall angle between j and E approaches 90°(j⊥E); and the current becomes tangent to the nanodots. This further indicates that the current is deflected to flow around the nanodots, resembling an "open circuit" state (Figure S5b in Additional file 1) [⁴²]. The transition from the "short circuit" at the zero field to the "open circuit" at high fields produces an increase of resistance, i.e., a positive geometrically-enhanced MR [⁴¹]. The above explanation has been successfully applied to several material systems, including Au/InSb [⁴¹] and MnAs/MnGaAs [⁴²]. Similarly, the geometrically-enhanced MR (ca. 200% at 10 Tesla, 300 K) was identified in MnGe₂ nanostructures with a high Mn concentration of approximately 33% [¹⁹].