1. Introduction

When Moore's law is approaching its limitation, the scientific community and industry want to extend its validity. A possible solution for extending validity is integrating magnetism into semiconductor materials, so electron and spin can be controlled at the same time. These are diluted magnetic semiconductor materials (DMSs). A natural idea is doping transitional metal ions (Fe, Co, Ni) into a semiconductor to form DMSs. Dietl et al. have proposed a theoretical model for this idea^[1]. This model needs charge carriers (electron or hole) to establish magnetic coupling between the metal ions in the semiconductor. However, the problem of this model is that transitional metal ions have limited solubility, so the Curie temperature ( $T_{\mathrm{c}})$ is lower than 300 K^[2]. Later, a new high T_c magnetic semiconductor was found, whose magnetism is induced by defects. This is so called d⁰ magnetism since no magnetic ion is included in this magnetism. In 2004, Coey et al. reported that HfO₂ can be ferromagnetic without doping any 3d elements^[3]. This discovery has challenged the traditional idea about the magnetism since neither Hf⁴⁺ and O^2- nor are magnetic ions and the d shells or f shells are either empty or full. So the possible reason for induced magnetism is the oxygen vacancies, namely defects. Very quickly this phenomenon was also discovered in the following structure materials, hex-borides, oxides^[4], nitrides and carbon-based materials, such as graphite and SiC^{[5, 6]}.

The diluted magnetic semiconductor and d⁰ magnetic materials have many aspects that can be compared. Taking SiC as an example, both materials have similar magnetization saturation (Ms), 10^-4 to 10^-6 emu/mg^[7]. For fabricating diluted magnetic SiC, many thin film growth methods can be used, such as ion implantation^{[2, 8]}, solid-state reaction^{[9, 10]}, chemical vapor deposition (CVD) and molecular beam epitaxy (MBE)^[11]. Due to the requirement of purity, implantation and neutron irradiation are the main methods for manufacturing d⁰ magnetic SiC. Excluding the secondary phase is important for both systems, since the secondary phase is usually composed of a transitional ion cluster which leads to high Curie temperature and misleads the understanding of magnetic mechanism for both systems. In general, X-ray diffraction (XRD) will be used for tough testing for the secondary phase^[11]. Then precise methods can be used for trace level detection, such as, particle-induced X-ray emission (PIXE) or synchrotron radiation tests^[10-12]. The concerns of the two systems are not exactly the same. Still using SiC as an example, Fe and Mn doped in SiC are the most researched SiC-based DMSs^[13-15]. The researchers focus on increasing hole concentration in order to fabricate practical DMSs. Most reports of, T_c of d⁰ magnetic SiC are already higher than 300 K. So researchers care more about magnetic properties about d⁰ magnetism, such as magnetic defect types, coupling mechanism, thermal stability etc.^[16]. In the near future those two methods of inducing magnetism could mix together after having more knowledge of d⁰ magnetism. Because the defect is ineluctable in film growth, defect-induced magnetism could be a possible solution for high T_c problem in DMSs.

This short review focuses on presenting our recent research progress of d⁰ magnetism in SiC. Some important experimental results are presented to describe a general idea about the status of d⁰ magnetism in SiC. Meanwhile, complex magnetic theory and sophisticated experimental details can be found in corresponding references. Previous reviews of d⁰ magnetism were done by Coey and Esquinazi in 2004^{[17, 18]}. Another purpose of this review is to attract the attention of a researcher who is working in diluted magnetic semiconductors by doping 3d elements, since the defects could be an important factor for further strengthening magnetic coupling, which has just recently been noticed.

In this review, several important works of d⁰ magnetism in graphite need to be briefly summarized before turning to introduce d⁰ magnetism in SiC, which aims for introducing the common methods of studying d⁰ magnetism in carbon-based materials. Then d⁰ magnetism in SiC is organized in two aspects, magnetic source and magnetic coupling mechanism. They reveal the different prospects for understanding d⁰ magnetism in SiC.

2. d⁰ magnetism in graphite

In 2003 Esquinazi et al. found that the diamagnetic graphite changed to be ferromagnetic after 2.2 MeV proton irradiation^[5]. The magnetic source is vacancy and d⁰ magnetism can be reproducible, and the Ms varied with the ion fluence. They also found other ferromagnetic properties, such as a weak magnetic anisotropy at low temperature, high $T_{\mathrm{c}}$ (over 380 K) and the domain structure. Assuming the magnetic region only happened in the irradiated area, the Ms is about 1.1 emu/g, which is still weaker than iron oxide (Fe₃O₄: 104.6 emu/g), but much stronger than other diluted magnetic materials. After this experiment, they also found that T_c can be influenced by the implanted fluence and the longitudinal magnetoresistance is irreversible^[6]. Meanwhile, the types of ion of induced ferromagnetism were also investigated^[4]. N⁺, C⁴⁺ and H⁺ were implanted to graphite with different fluences at energy around MeV. The results showed that only a few set parameter combinations can induce ferromagnetism. The probable reason is that the distance between vacancies is too long to block the ferromagnetic coupling. In the year of 2010, their X-ray magnetic circular dichroism (XMCD) test showed that the most magnetic signals come from carbon in the surface region with depth of 10 nm, so it means that the Ms of effective region is nearly 25% about Ni^[19]. The possible reason for ferromagnetic coupling is the formation of spin-polarized C-H bond in the process of irradiating graphite. It also explains the reason why further increasing fluence will damage the magnetization. Besides searching for the source of magnetic coupling, the magnetic elements contamination in graphite is also the key problem for explaining d⁰ magnetism in graphite since the pure graphite without contamination is hard to obtain in experiment. A few magnetic element clusters could often be found in the boundaries of graphite layers^[20]. In 2014, the group systemically tested the content of tracing 3d elements in graphite of main suppliers with Particle-induced X-ray Emission (PIXE) and Rutherford Backscattering Spectrometry Channeling (RBS-C). The results revealed that the influence of the 3d element can be ignored after carefully cleaning^[21].

Although existence of d⁰ magnetism in graphite has been systematically tested, the magnetic saturation of graphite is hard to further increase, which greatly limits the further applications. The precise control of defect distribution in graphite may be the key issue for the next research.

3. d⁰ magnetism in SiC

3.1 Why SiC

The research of d⁰ magnetism in graphite is still under way. As engineering materials, d⁰ ferromagnetism in graphite can have applications in electromagnets. But for microelectronic industry, a demand is that d⁰ magnetism can be integrated into the Si wafer. So after researching ZnO^{[4, 22, 23]}, MoS₂^[24] and graphite, SiC is followed. Comparing with the graphite, SiC has several advantages^[25-27]. (1) In SiC, the 3d element concentration is in the range of 10¹⁴ atoms/cm³ (0.015 μg/g), which is one order of magnitude less than high pure graphite sample^[28]. (2) SiC is an important material for the microelectronic industry, which is more compatible with Si-based technology than other materials. (3) SiC contains p electrons from C atoms, so ferromagnetism is possibly induced by defects.

3.2 Divacancy in SiC

The first round of reports of d⁰ magnetism in SiC at room temperature began from 2011 by ion implantation and neutron irradiation (see Fig. 1)^{[29, 30]}. The type and concentration of magnetic defects in the two kinds of samples were tested by positron annihilation lifetime spectroscopy (PALS). The life time is 239 $\pm$ 20 ps which is independent on the fluence^[31]. The defect probably is divacancy which also possibly is the source of ferromagnetism^{[32, 33]}. So a short review of spin states of divacancy is necessary.

Divacancies in 4H-SiC and 6H-SiC are formed by the combination of nearest single Si vacancy and single C vacancy, with two different positions which correspond to P6/P7 in electron paramagnetic resonance (see Fig. 2)^{[34, 35]}. Son showed that P6/P7 are easier to detect in 4H-SiC when the SiC is irradiated by 3 MeV electrons with the fluence over 1.1 × 10¹⁹ cm^-2. The P6 peak (axial configuration) can be found by the magnetic field perpendicular to the ab plane and P7 peak can be observed by the magnetic field in the direction deviated by 70° from c axial. Their ab initio supercell calculations revealed that a high spin state is the ground state of the neutral divacancy with spin S=1. The spin density is mainly distributed on the nearest carbon atoms, namely V_Si provides most spin density whereas the contribution of V_C can be negligible. So each neutral divacancy can provide a magnetic moment of 2 $\mu_{\mathrm{B}}$ . Some other calculation also showed similar results^{[36, 37]}, however so far 2 $\mu_{\mathrm{B}}$ provided by each divacancy is not tested by experiment. Another distinct character of divacancy is high thermal stability. Divacancy disappears when annealed at 1300 ℃^[38]. This is one of the possible reasons for room temperature Tc of d⁰ magnetic SiC.

Now divacancy is a source of d⁰ ferromagnetism in SiC, but a divacancy contains C and Si elements, so the weight of spin polarization is a problem. XMCD is an important element-specific experimental technique for identifying magnetism source, which can test the magnetic contribution of specific element in a magnetic sample. When the magnetic contribution of the specific element is strong, clear and sharp peaks will appear in XMCD spectra, and the spin and orbital contribution of this element can be calculated by so-called sum rule^[39]. However, for all the carbon-based materials this calculation is unpractical because of two weaknesses. First, the X-ray beamline usually contains some carbon contamination such as CO₂ which is hard to clean up, so the signal will include a background. Second, for the carbon the XMCD can only test the orbital contribution^{[19, 40-42]}.

Wang et al. measured K-edge of Si (1848 eV) and C (284 eV) of ferromagnetic sample (5E12) and pristine sample, respectively. The results are shown in Fig. 3 ^[43].

Figs. 3(a) and 3(c) are X-ray absorption spectra (XAS) of ferromagnetic sample and pristine sample for silicon K-edge and carbon K-edge, while Figs. 3(b) and 3(d) are their corresponding XMCD spectra. The XMCD signal of ferromagnetic sample and pristine sample are overlapped at silicon K-edge, so the silicon dangling bond cannot induce spin-polarized states.

On the contrary, the carbon K-edge XAS of two samples are quite different. A clear XMCD signal appears from 284 to 295 eV as shown in Fig. 3(d). In Fig. 3(c), for both samples, peaks around 285 and 290 eV correspond to $\pi$ ^* and σ^* resonances, respectively. The peak shape of ferromagnetic sample at 285 eV is quite like the graphitic materials which is sharper than the pristine sample^[44]. It indicates that sp³-type carbon in SiC is transformed to graphitic sp²-type carbon. The orthogonal pz orbital of sp³-type is unchanged in order to raise the peak of $\pi$ ^* bands in sp²-type. So the probable source of d⁰ magnetism in defective SiC is the unchanged pz orbital, which induces a spin-polarized state in carbons of divacancy.

3.3 Research of magnetic coupling

After the discussion of some important experimental results about the source of ferromagnetism, the next is the coupling mechanism of d⁰ magnetism in SiC by investigating the magnetic interaction region. In Fig. 1 not all the samples generate ferromagnetic coupling and the Ms is around 10^-3 emu/g which is not strong. There is some special fluence region where the ferromagnetism can be induced by defects, in which the special fluence range can be visually called a “Window”. The similar situation also happened in the ion-implanted case^[45].

Fig. 4(a) is the results from superconducting quantum interference device (SQUID) measurements of the hysteresis loop of series Ne implanted 6H-SiC without the diamagnetic background. The Ms also varies with the magnetic fluence. The maximum of Ms happens at fluence of 1 × 10¹⁴ cm^-2, then the Ms drops rapidly with further increasing fluence. The corresponding structure change is tested by XRD. The result is shown in Fig. 4(b), in which the big peak comes from the substrate and the satellite peaks shift to the left with the increasing fluence. It is indicated that the distance between ab planes of SiC diminishes and more defect clusters are created with fluence rising. Ms and strain level from XRD are summarized by a parameter dpa (displace per atom), which is used to characterize amorphization level^[46]. The small dpa value means light damage. The maximum Ms happens at shadow region in which the strain level is quite low.

Now it is clear that the d⁰ magnetism only happens in certain fluence region in which the damage of SiC is small. The ion implantation can only influence the surface layer (around 200 nm depth). A very natural idea is whether this defect will be induced into the whole SiC to obtain stronger Ms. Based on this idea, Yutian Wang performed the neutron irradiated experiment with fluences from 10¹⁶ n/cm² (low fluence) to 10¹⁹ n/cm² (high fluence)^[47]. When the fluence reaches 10¹⁹ n/cm² the sample approaches amorphization, and all the defects are probably divacancies (Fig. 5(a)). However, most of the samples are paramagnetic (Fig. 5(b)). Only a few low fluence irradiated samples showed ferromagnetic properties. To re-check the low fluence cases, the highest fluence sample was processed with isochronal annealing at 900 ℃ for testing the magnetic properties again. Some annealed samples were found with ferromagnetic loops as expected. But the Ms of annealed sample is not much stronger than the ferromagnetic samples without annealing (Fig. 6).

So it is not an effective way for strengthening ferromagnetic coupling to only increase density of divacancy. The inspiration of annealing is the structure of ferromagnetic samples should have some comparability. In other words, all of the magnetic samples should have similar structure no matter whether annealing or not. According to the ion cases, all the ferromagnetic samples occurred in the relatively low damage region, but structural characterized methods are not suitable for the neutron case since the penetration depth of neutron is in centimeter level. The parameters for describing bulk damage are necessary. The paramagnetic centers from fitting MH curve by Brillouin function (1) can be used for quantitatively expressing damage states of SiC.

$\begin{align} & M(x)= \\ & NJ{{\mu }_{\text{B}}}{{g}_{\text{J}}}\left[ \frac{2J+1}{2J}\coth \left( \frac{2J+1}{2J}x \right)-\frac{1}{2J}\coth \left( \frac{1}{2J}x \right) \right]. \\ \end{align}$

(1)

In this way all the samples can be plotted in the same picture with change of neutron fluence. As shown in Fig. 7, the red circle (solid circle) stands for the ferromagnetic sample at low fluence, and the blue triangle (solid upper) means the annealed ferromagnetic samples. It can be clearly seen that all the ferromagnetic samples are still in the narrow window region which illustrates all the FM samples have similar paramagnetic background. This proved the fact again: the ferromagnetism only happens in a special region.

Now another question arises: how do those regions interact and form the ferromagnetic coupling in SiC? To answer this question, it is necessary to identify how many magnetic phases are in d⁰ SiC^[16]. Let us go back to the ion implanted case. Intrinsic SiC is diamagnetic at 300 K, however the intrinsic defect in SiC exhibits the quite strong paramagnetic property at 5 K. As shown in Fig. 8(a), the red and blue curves (circle and upper triangle) are MH of pristine sample at 5 and 4.2 K, and the obvious arc deviation is in paramagnetic background which is probably from paramagnetic point defect in the substrate. So the d⁰ magnetic SiC prepared by ion implantation should have ferromagnetic, diamagnetic and paramagnetic phases which are from divacancy, substrate, and point defect created by the process of implantation (Fig. 8(b)). To search other magnetic phases, the mentioned contributions should be removed one-by-one. Fig. 8(c) shows first quadrant of process of disentangling d⁰ magnetism, and the red curve with solid square means after subtracting the diamagnetism and paramagnetism from the pristine sample. Still some paramagnetism remains in the matrix.

$M={{M}_{\text{s}}}(0)[1-{{(T/{{T}_{\text{c}}})}^{3/2}}]+n\mu [\coth (\mu H/{{k}_{\text{B}}}T)-{{k}_{\text{B}}}T/\mu H],$

(2)

This kind of contribution could be paramagnetic which is generated in the process of ion implantation and also could be the superparamagnetism. For identifying the contribution, zero cooling field and field cooling (ZFC-FC) test was adopted for testing the historical effect of the field. The branching shape clearly shows that the contribution around 5 K should be superparamagnetism. However, a similar experiment has been done in the neutron case but no branching shape occurred. Temperature-dependent magnetic remanence is tested and analyzed by Eq. (2). The first part of the function stands for the contribution of ferromagnetism and the second stands for the superparamagnetism. The system can be treated as a sum of both contributions. The model fitted the experimental data well and the results are listed in Table 1.

From Table 1, the fitted Tc is 762 K, and Ms(0) is ferromagnetic contribution. However, total magnetic moment of superparamagnetism and the magnitude approaches the ferromagnetic contribution if n multiplied by μ is 10¹⁸μB/g. One cluster contains about 4000 divacancies which is quite a large region in the SiC if assuming each divacancy contains 2μB. The reason for forming superparamagnetism is still not clear, but the coexistence of ferromagnetism and superparamagnetism in the same matrix indicates that the distance between the divacancies may be the key point. In other words d⁰ magnetism can be explained by a cluster model and assuming that magnetic divacancies can form the superparamagnetic clusters, then superparamagnetic clusters can be of ferromagnetic coupling when the distance between them has satisfied the critical interval value.

3.4 Theoretical research of d⁰ ferromagnetism in SiC

The theoretical investigations also began when the d⁰ ferromagnetism was found in SiC. In those investigations, most of them are based on the first-principle calculation. Liu et al. researched the coupling mechanism according to the nearest-neighbor Heisenberg model. They made a SiC supercell model which contains a spin-polarized V_Si-V_C (as shown in Fig. 10 left picture). Then they combine the two same supercells into a big new single supercell which is shown Fig. 10. So now this new one has two spin-polarized V_SiV_C. So the two spin-polarized divacancies can be ferromagnetic coupling (FM) when the spin-polarized directions are the same; contrariwise it is antiferromagnetism (AFM) when the spin-polarized directions are opposite. Then the energy difference between the FM and AFM was calculated for determining energy stable states. They also investigated changed states, and found changed states like (-1, -1) can form ferromagnetic coupling and unchanged states and (+1, +1) can form antiferromagnetic coupling. In 2015 they promoted the model. The relationship between the distance of two coupling divacancies and the energy difference of AFM and FM were carefully discussed^[47]. Two cases were considered in this model. Divacancies coupling happened along crystallographic axes a and c, respectively. For both cases, no coupling happened when the distance was smaller than 10.11 Å. The unchanged states and (+1, +1) states still form the AFM. The coupling strength decreases rapidly with distance increasing.

The function of the other atom in d⁰ ferromagnetic SiC was also discussed by first-principle calculation. In 2013, He et al. discussed divacancy coupling model under the circumstance of varying-concentration hydrogenation. So the model can be written as [V_SiV_C + nH, (n =0-3)]. Note that the ferromagnetic coupling is enhanced when the model hydrogenates only one H atom. Possibly, in d⁰ ferromagnetic system, the effect of the other atom to coupling, such as H, is weak but important, which has been reported in d⁰ magnetic graphite^{[19, 48]}. Besides the divacancy model, Zheng et al.^[49] proposed the Cu_si+V_si model, since they realized d⁰ ferromagnetism by implanted copper in SiC. Their method is similar to the divacancy model. Their DOS (density of states) analysis shows that the carbon P electron has a dominant effect. However, the Cu d DOS is non-zero at Fermi-level, even though it is very weak. So more experiments are needed to explain how does the Cu join the coupling.

In the future, simulation will play a more important role in the research of d⁰ magnetism in SiC or other materials. With increasing experimental results, the large-scale method is urgently needed. In particular, it is required that the method can investigate the magnetization under various temperatures. So combination of other large-scale methods, such as Monte Carlo and molecular dynamics could be the next step for theoretical research.

Now from reported experiments we can further summarize some ideas for the d⁰ magnetism in SiC. Both the neutron and ion cases show that the ferromagnetism appears in the certain region where the damage is quite low. Especially in the ion case, the superparamagnetic phase and ferromagnetic phase can coexist in the same matrix. This means that the d⁰ magnetism is a kind of localized effect where the distance between the divacancies and distribution of defects has to satisfy a certain limit. For the certain limit, Yuliu et al calculated the distance of two divacancies by first-principles approach. The results showed that the ferromagnetic coupling distance is around 1 nm, which somehow evidenced that the distance is quite important for defect-induced ferromagnetism, but first-principles calculation still cannot examine divacancy and cluster distribution because of limitation of calculation capability.

4. Discussion

So far the experimental evidence shows us a scenario of formation of defective ferromagnetism in SiC. The ion implantation generates defect regions. In some special region, where the divacancies have high density, the ferromagnetic coupling or superparamagnetic coupling below the $T_{\mathrm{c}}$ or block temperature ( $T_{\mathrm{B}})$ can be formed respectively. To further promote the topic, several following problems should be considered at first.

The first issue is how to make d⁰ ferromagnetism in SiC mass reproducible.

From 2011 to 2016, the d⁰ in SiC is continuously reported by different groups, but no one systematically studied the condition of mass reproduction of d⁰ in SiC. However, before advanced spacing technique can directly detect the coupling between the two divacancies, it is a practical way to produce massive d⁰ samples to further find the mechanism and promote the application.

The second issue is how the divacancies or more defects make ferromagnetic coupling?

Although most of the research showed that the divacancy probably is the source of d⁰ ferromagnetism, and the electron spin resonance(ESR) and positron annihilation lifetime spectroscopy (PALS) had also demonstrated the neutral divacancy can induce a polarized state in the SiC, no experiments support that divacancies can directly cause ferromagnetic coupling. So more efforts are also expected. Meanwhile, other defects, which can also induce spin polarization, should also be noticed, such as V_Si^[50].

The third issue is the function of paramagnetic centers.

The function of paramagnetic defects is ignored in the research of ion-implanted samples. Samples had a similar paramagnetic background until the differences in all the neutron-irradiated samples were found. Comparing with the divacancy, the paramagnetic centers are composed of point defects whose thermal stability is low but has strong movability. The point defect, special paramagnetic defect, can be the magnetic messenger to deliver the magnetic information between the divacancies.

The fourth is the practical applications, i.e., could d⁰ magnetism be possibly combined with 3d elements in the same system?

To strengthen Ms, d⁰ magnetism can be associated with diluted ferromagnetic materials, such as Mn-doped SiC. Wang et al. found the Mn-doped SiC can have room temperature T_c, but the T_c of ferromagnetic Mn₄Si₇ is only 44 K^[41]. So the only possible reason is the defect generated in the process of fabrication. Coey had also mentioned the possible application for d⁰ is how to send the d⁰ to the certain position. From the investigation of the accompanying defect in the Mn implanted process, there may be a new relation between the magnetic defect and 3d elements.

At last, SiC-based nano-size materials will be the next d⁰ ferromagnetic material. In 2010 the magnetic properties of semihydrogenated SiC sheet was discussed with density functional theory (DFT) method. The calculated results showed that hydrogenated Si atoms lead a ferromagnetic semiconductor, while hydrogenated C atoms lead an antiferromagnetic semiconductor^[51]. In 2011, Zhang et al. studied d⁰ ferromagnetism in single-wall SiC nanotubes by density functional theory (DFT) methods^[52]. In armchair (4, 4) nanotube, a C vacancy, the magnetic moment could only be brought about.

Many experimental results are also reported about nano-scale d⁰ magnetic SiC-based structure. Mishra et al. produced ferromagnetic nanocrystalline beta-SiC by ball-mill^[53]. Four-hour ball-milled beta-SiC product synthesized by the thermal plasma technique showed room temperature ferromagnetism. The semi-lambda signature of the field-cooled magnetization (FC) and zero field-cooled magnetization (ZFC) curves suggest the possible signature of a glassy ferromagnetism state in the sample. The prominent fall in the magnetization value at around 50 K observed in ZFC curve reveals the existence of a sharp transition from a ferromagnetic state to a glassy ferromagnetic state. The presence of glassy ferromagnetism at low temperature is confirmed from the M-H curve recorded at 5 K. So we can predict that d⁰ probably finds application in the nano-scale materials in the near future.

5. Conclusions

In this article, our research of defect-induced ferromagnetism in SiC has been reported. The global progress of d⁰ in carbon-based materials has also been briefly reviewed. In SiC, the structural change with irradiated fluences in the ferromagnetic sample has been investigated. The carbon p electrons in divacancy are the source of ferromagnetism, which are tested by XMCD. Finally, the magnetic coupling mechanism is also studied. Those results support that the defect-induced ferromagnetism can be treated as a kind of local effect. However, it is hard to say the effect can be a bulk effect before revealing the magnetic coupling mechanism. According to recent knowledge, the defect may also play an important key role in 3d element doped SiC. It is the possible solution to combine the diluted magnetism and d⁰ magnetism in SiC for breaking the deadlock in these two fields.