Interstitial Atom Engineering in Magnetic Materials

: Interstitial light elements play an important role in magnetic materials by improving the magnetic properties through changes of the unit cell volume or through orbital hybridization between the magnetic and interstitial atoms. In this review focusing on the e ﬀ ects of interstitial atoms in Mn-based compounds, which are not well researched, the studies of interstitial atoms in three kinds of magnetic materials (rare-earth Fe-, Mn-, and rare-earth-based compounds) are surveyed. The prominent features of Mn-based compounds are interstitial-atom-induced changes or additional formation of magnetism—either a change from antiferromagnetism (paramagnetism) to ferromagnetism or an additional formation of ferromagnetism. It is noted that in some cases, ferromagnetic coupling can be abruptly caused by a small number of interstitial atoms, which has been overlooked in previous research on rare-earth Fe-based compounds. We also present candidates of Mn compounds, which enable changes of the magnetic state. The Mn-based compounds are particularly important for the easy fabrication of highly functional magnetic devices, as they allow on-demand control of magnetism without causing a large lattice mismatch, among other advantages.


Introduction
Some crystal structures possess interstitial crystallographic sites, which light elements such as hydrogen, boron, carbon, nitrogen, and oxygen atoms can occupy. There is a rather long history of metallurgical, physical, and chemical research studies on interstitial atoms [1,2]. In past years, domain control of ferromagnets has been studied [1]. Since the 1980s, interstitial atoms have attracted intense attention related to the improvement of magnetic properties of rare-earth Fe and Co-based permanent magnets [2][3][4][5][6].
Interstitial atoms have two major roles-influencing the stability of the crystal structure and in the modification or change in magnetic properties. In the former case, small amounts of interstitial light elements are required to stabilize the desired crystal structure; compounds without interstitial atoms would not exist in thermal equilibrium. In the latter case, interstitial atoms affect the crystal structure parameters, such as the interatomic distances between magnetic atoms or the orbital hybridization between magnetic and interstitial atoms, consequently meaning the magnetic ordering temperature, the magnetic moment, the magnetic structure, and other factors can be altered.
The most well-studied platforms for interstitial atoms are rare-earth Fe-based permanent magnets. The improvements of the magnetic properties have mainly been achieved through the addition of light On the other hand, the effects of interstitial atoms in Mn-based compounds are not well researched. As shown in Figure 1, the Mn atom itself shows the AFM ground state, however the expanded Mn-Mn distance in Mn-based compounds leads to the FM state. Mn compounds are indispensable for both FM and AFM materials. For example, MnBi and MnAl have attracted much attention as permanent magnets [14][15][16][17][18]. MnSi has been extensively studied as a magnetic material with a skyrmion state, which is a noncollinear magnetic structure. The skyrmion domains can be driven by the low current density threshold [19]. Recently, Mn3Sn has been intensively studied as a topological antiferromagnet [20] and is a candidate next-generation spintronics material. If the effects of interstitial atoms are well understood, they can be highly useful in the development of spintronics devices or highly functional magnetic devices, which can be developed via easy on-demand control of the magnetic state.
In this review, we present information on the effects of interstitial atoms in rare-earth Fe-, Mn-, and rare-earth-based compounds, especially focusing on Mn-based compounds. We introduce the frequently studied Mn-based compounds, showing the changes of magnetic states caused by the interstitial atoms, as well as our recent results. Interestingly, a change in the magnetic ground state or an additional formation of the FM state caused by interstitial atoms is possible. Specifically, our results highlight an abrupt emergence of FM exchange coupling above room temperature caused by the addition of a small amount of light elements, which has not been clearly reported for rare-earth Fe-based compounds. Additionally, candidate crystal structures for Mn compounds are presented, in which a magnetic state can be changed using interstitial atoms. Finally, perspectives concerning the development of magnetic devices based on Mn-based compounds and the strategy for further improvement of magnetic properties are outlined. On the other hand, the effects of interstitial atoms in Mn-based compounds are not well researched. As shown in Figure 1, the Mn atom itself shows the AFM ground state, however the expanded Mn-Mn distance in Mn-based compounds leads to the FM state. Mn compounds are indispensable for both FM and AFM materials. For example, MnBi and MnAl have attracted much attention as permanent magnets [14][15][16][17][18]. MnSi has been extensively studied as a magnetic material with a skyrmion state, which is a noncollinear magnetic structure. The skyrmion domains can be driven by the low current density threshold [19]. Recently, Mn 3 Sn has been intensively studied as a topological antiferromagnet [20] and is a candidate next-generation spintronics material. If the effects of interstitial atoms are well understood, they can be highly useful in the development of spintronics devices or highly functional magnetic devices, which can be developed via easy on-demand control of the magnetic state.

Th2Zn17-Type
In this review, we present information on the effects of interstitial atoms in rare-earth Fe-, Mn-, and rare-earth-based compounds, especially focusing on Mn-based compounds. We introduce the frequently studied Mn-based compounds, showing the changes of magnetic states caused by the interstitial atoms, as well as our recent results. Interestingly, a change in the magnetic ground state or an additional formation of the FM state caused by interstitial atoms is possible. Specifically, our results highlight an abrupt emergence of FM exchange coupling above room temperature caused by the addition of a small amount of light elements, which has not been clearly reported for rare-earth Fe-based compounds. Additionally, candidate crystal structures for Mn compounds are presented, in which a magnetic state can be changed using interstitial atoms. Finally, perspectives concerning the development of magnetic devices based on Mn-based compounds and the strategy for further improvement of magnetic properties are outlined. Table 1. The magnetic and structural properties of rare-earth Fe-based compounds with and without interstitial atoms, where a and c are the lattice parameters, and V is the unit cell volume. RT means the room temperature. Data are from the references listed in the table.
Compound The addition of nitrogen atoms expands the lattice parameters a and c, resulting in the increased Fe-Fe interatomic distance and the enhancement of T C . Actually, in Sm 2 Fe 17 N 3 , T C is almost doubled, which is much larger than that in the case of ThMn 12 -type compounds (see also Section 2.2 and Table 1). The difference is well correlated with the rate of unit-cell-volume change ∆V/V by the interstitial atoms, which is defined by where V w/o and V w are the unit cell volumes without and with interstitial atoms, respectively. We note here that the compound without interstitial atoms is hereafter often called the parent compound. For example, ∆V/V = 6.6% in Sm 2 Fe 17 N 3 is larger than that in ThMn 12 -type SmFe 11 TiN (2.7%). The larger ∆V/V for Sm 2 Fe 17 N 3 is ascribed to the fact that Fe magnetic moments in the Th 2 Zn 17 -type structure are less itinerant.

ThMn 12 -Type
The well studied system is RFe 11 Ti (R: rare-earth) series (I4/mmm, No. 139). This structure allows the interstitial nitrogen atoms to occupy the 2b site. In SmFe 11 TiN or NdFe 11 TiN, the nitrogen addition enhances T C by about 30%, which is smaller than that of the Th 2 Zn 17 -type system (see Table 1). As mentioned above this is due to the smaller ∆V/V under the addition of nitrogen (2.7% for SmFe 11 TiN and 3.9% for NdFe 11 TiN). The magnetic anisotropy of RFe 11 Ti is determined primarily due to the single-ion contribution of R ion, thus NdFe 11 Ti and SmFe 11 Ti show the planar and c axis anisotropies, respectively. In each compound, the sign of magnetic anisotropy constant K 1 is reversed by the nitrogen addition and a good candidate of the permanent magnet is consequently NdFe 11 TiN.
The first-principle studies have revealed that NdFe 12 N, which possesses an Fe concentration higher than that of NdFe 11 TiN and is more favorable for a permanent magnet, has excellent magnetic properties [35][36][37], however, it is thermodynamically unstable in bulk form. The thin film fabrication allows the growth of NdFe 12 N x , which shows a superior M s = 1.7 T compared to Nd-Fe-B magnets [38]. The coercivity of NdFe 12 N x is rather low, however, a born doped Sm(Fe 0.8 Co 0.2 ) 12 is recently reported to be a highly coercive material with 1.2 T [39]. The microstructure of the compound exhibits columnar-shaped Sm(Fe 0.8 Co 0.2 ) 12 grains, surrounded by a born-rich amorphous phase. The domain wall pinning at the grain boundary is responsible for the high coercivity [39]. Mao et al. reported a BH energy product of ThMn 12 -type compound [40]. They have investigated the magnetic properties of powdered PrFe 12-x V x and the nitride compound. Pr(Fe, V) 12 N 1.6 exhibits the maximum BH energy product of 135 kJ/m 3 .

BaCd 11 -Type
The tetragonal BaCd 11 -type structure represented by RFe 9 Si 2 C x (Table 1) is a candidate for a next-generation rare-earth Fe-based permanent magnet. It is interesting that the addition of carbon atoms is required to stabilize the BaCd 11 -type structure. In RFe 9 Si 2 C x , the carbon atom occupies the interstitial 8c site of the tetragonal space group of I4 1 /amd (No. 141). T C progressively increases with the increase of x, for example, T C = 367 K of SmFe 9 Si 2 C 0.5 is enhanced to 492 K in SmFe 9 Si 2 C 1.5 , accompanying the volume expansion [30]. The magnetic anisotropy depends on the R atom; in-plane anisotropy is observed in R = Ce or Nd, whilst on the other hand, R = Sm possesses the c axis anisotropy.

Remark on Hybridization Effect
Through the extensive studies of rare-earth Fe-based permanent magnets, it is revealed that stronger orbital hybridization in the covalent bond between Fe and an interstitial atom tends to reduce the magnetic moment. The strength of the covalent bond is well reflected by ∆V/V under the addition of light elements; ∆V/V is not so large when the hybridization is stronger. ∆V/V is generally larger for the nitrogen addition compared to the carbon one (see, for example, Sm 2 Fe 17 N 3 and Sm 2 Fe 17 C 3 in Table 1).

Mn-Based Compounds
In this section, we gathered as many Mn-based room temperature ferromagnets as possible.
3.1. Hydrogen-Absorbed (R or Th) 6 Mn 23 R 6 Mn 23 (R = rare earth) series and Th 6 Mn 23 crystallize into the cubic Th 6 Mn 23 -type structure with the space group Fm3m (No. 225). R atoms occupy the 24e site, and Mn atoms the 4b, the 24d and the two 32f sites. All R 6 Mn 23 compounds show FM orderings at a T C higher than 300 K, which are induced by Mn magnetic moments. On the other hand, Th 6 Mn 23 remains a paramagnet down to low temperatures. The hydrogen absorption generally expands the unit cell volume (see Figure 2 and Table 4), whereas the occupation site of the hydrogen atom is not clear. Although the T C of non-hydrogenated R 6 Mn 23 rises with increasing volume, most of the hydrogen-absorbed compound shows the suppressed T C compared to each parent compound. For example [41], T C = 461 K of Gd 6 Mn 23 is reduced to T C = 2.66 K in Gd 6 Mn 23 H x . The saturated moment is also highly reduced (49 µ B /f.u. to 14.2 µ B /f.u.) [41]. The opposite behavior is confirmed for Th 6 Mn 23 , in spite of the similar volume change by the hydrogenation. The hydrogenated Th 6 Mn 23 shows an emergence of ferromagnetism with T C = 335 K, although the saturated moment of Mn is not so high (16.5 µ B /f.u.) [41]. Considering that R and Th ions are in the trivalent and the tetravalent state, respectively, the exchange coupling between Mn magnetic moments would significantly depend on the valence electron count per atom (VEC). We note that Th 6 Mn 23 is the typical Mn-based compound showing the change in magnetic state by interstitial atoms. The saturated moment induced in the FM state of the hydrogenated compound is rather low, indicating a strong orbital hybridization between Mn and hydrogen atoms and/or a complex magnetic structure.

Hydrogen-Absorbed YMn 2
The crystal structure of YMn 2 is the cubic MgCu 2 -type structure with the space group of Fd3m (No. 227). Y and Mn atoms occupy the 8a and the 16d site, respectively. YMn 2 can absorb hydrogen atoms as in R 6 Mn 23 . Although YMn 2 is paramagnetic down to low temperatures, YMn 2 H x shows a lattice expansion under hydrogenation, which induces a ferromagnetism with T C = 284 K and a saturation moment of 0.52 µ B /f.u. [41]. There are many RMn 2 compounds with the same structure, however, we do not make the magnetic ordering temperature vs. unit cell volume plot, because the accurate nature of the magnetic order is still unknown for each compound [42]. YMn 2 can also be regarded as the compound, realizing the change from paramagnetism to ferromagnetism by interstitial atoms.

Hydrogen-Absorbed YMn2
The crystal structure of YMn2 is the cubic MgCu2-type structure with the space group of Fd3 m (No. 227). Y and Mn atoms occupy the 8a and the 16d site, respectively. YMn2 can absorb hydrogen atoms as in R6Mn23. Although YMn2 is paramagnetic down to low temperatures, YMn2Hx shows a lattice expansion under hydrogenation, which induces a ferromagnetism with TC = 284 K and a saturation moment of 0.52 µB/f.u. [41]. There are many RMn2 compounds with the same structure, however, we do not make the magnetic ordering temperature vs. unit cell volume plot, because the accurate nature of the magnetic order is still unknown for each compound [42]. YMn2 can also be regarded as the compound, realizing the change from paramagnetism to ferromagnetism by interstitial atoms.

Carbon-Added Mn5Si3
This system shows a change from the AFM to FM state by the carbon addition. The hexagonal Mn5Si3-type structure is known as a superior platform for studying interstitial chemistry [43]. The space group is P63/mcm (No. 193), in which the Wyckoff positions are the 4d for Mn1, the 6g for Mn2 and the 6g for Si (see also Figure 3a). Carbon atoms occupy the 2b site. While the parent compound Mn5Si3 shows an AFM ordering at the Néel temperature TN = 98 K, a thin film Mn5Si3Cx becomes a room-temperature ferromagnet with TC = 350 K as shown in Figure 3b (see the filled squares) [44,45]. In [44], the appearance of the FM state is ascribed to the enhanced Mn-Mn interaction mediated by added carbon. Associated with the unit cell volume expansion by the carbon addition, the saturation Mn moment rapidly increases to 1 µB/Mn at x = 0.75 (see Figure 3c). However, the x dependence of TC is peculiar: a TC plateau of 350 K in a rather wide x range, where the saturation Mn moment is steadily reduced as x is increased above 0.75. We speculate that the hybridization between Mn2 and C atoms is strong, due to the short Mn2-C distance (see Figure 3a). As x is increased, the magnetic moment of Mn2 would be decreased, but that of Mn1 under well localized state due to the weak influence of the carbon addition would be enhanced. The rather strong Mn1-Mn1 magnetic interaction would be responsible for the plateau of TC, while the decreasing Mn2 moment with increasing x would contribute to the reduction in the saturation moment.

Carbon-Added Mn 5 Si 3
This system shows a change from the AFM to FM state by the carbon addition. The hexagonal Mn 5 Si 3 -type structure is known as a superior platform for studying interstitial chemistry [43]. The space group is P6 3 /mcm (No. 193), in which the Wyckoff positions are the 4d for Mn1, the 6g for Mn2 and the 6g for Si (see also Figure 3a). Carbon atoms occupy the 2b site. While the parent compound Mn 5 Si 3 shows an AFM ordering at the Néel temperature T N = 98 K, a thin film Mn 5 Si 3 C x becomes a room-temperature ferromagnet with T C = 350 K as shown in Figure 3b (see the filled squares) [44,45]. In [44], the appearance of the FM state is ascribed to the enhanced Mn-Mn interaction mediated by added carbon. Associated with the unit cell volume expansion by the carbon addition, the saturation Mn moment rapidly increases to 1 µ B /Mn at x = 0.75 (see Figure 3c). However, the x dependence of T C is peculiar: a T C plateau of 350 K in a rather wide x range, where the saturation Mn moment is steadily reduced as x is increased above 0.75. We speculate that the hybridization between Mn2 and C atoms is strong, due to the short Mn2-C distance (see Figure 3a). As x is increased, the magnetic moment of Mn2 would be decreased, but that of Mn1 under well localized state due to the weak influence of the carbon addition would be enhanced. The rather strong Mn1-Mn1 magnetic interaction would be responsible for the plateau of T C , while the decreasing Mn2 moment with increasing x would contribute to the reduction in the saturation moment.

(R or Actinide)Mn 2 Si 2 and Its Germanides
Despite the absence of a report on the addition of light elements in these compounds, the magnetic ordering temperature vs. unit cell volume plot shows the thought-provoking results as shown in Figure 4. These compounds possess the tetragonal ThCr 2 Si 2 -type structure (I4/mmm, No. 139), where Mn atoms at the 4d sites form the layered structure perpendicular to the c axis. R (Actinide) and Si (Ge) atoms occupy the 2a and the 4e sites, respectively. In the case of RMn 2 Si 2 , only the R = La compound exhibits an FM state and the other compounds AFM one (see 21-35 in Figure 4). When Si is replaced by Ge, FM (AFM) states are observed for R = La to Sm (R = the other elements) as shown in 36-47 of Figure 4. Figure 4 provides a good correlation between the magnetic ordering temperature and the unit cell volume throughout the two series: systematically descending ordering temperature with expanding volume. Furthermore, magnetic structure changes from AFM to FM at approximately 179 Å 3 , which is consistent with the picture of the Bethe-Slater curve. The inverse trend is confirmed in (U or Th)Mn 2 Si 2 and its germanide (see 48 and 49 (50 and 51) in Figure 4), that is, the crossover from the FM to AFM state occurs by increasing the volume in each system. This can be ascribed to the difference of valence between rare-earth (trivalent) and actinide (maybe tetravalent) elements as in the case of Th 6 Mn 23 -type compounds.

(R or Actinide)Mn2Si2 and Its Germanides
Despite the absence of a report on the addition of light elements in these compounds, the magnetic ordering temperature vs. unit cell volume plot shows the thought-provoking results as shown in Figure 4. These compounds possess the tetragonal ThCr2Si2-type structure (I4/mmm, No. 139), where Mn atoms at the 4d sites form the layered structure perpendicular to the c axis. R (Actinide) and Si (Ge) atoms occupy the 2a and the 4e sites, respectively. In the case of RMn2Si2, only the R = La compound exhibits an FM state and the other compounds AFM one (see 21-35 in Figure  4). When Si is replaced by Ge, FM (AFM) states are observed for R = La to Sm (R = the other elements) as shown in 36-47 of Figure 4. Figure 4 provides a good correlation between the magnetic ordering temperature and the unit cell volume throughout the two series: systematically descending ordering temperature with expanding volume. Furthermore, magnetic structure changes from AFM to FM at approximately 179 Å 3 , which is consistent with the picture of the Bethe-Slater curve. The inverse trend is confirmed in (U or Th)Mn2Si2 and its germanide (see 48 and 49 (50 and 51) in Figure 4), that is, the crossover from the FM to AFM state occurs by increasing the volume in each system. This can be ascribed to the difference of valence between rare-earth (trivalent) and actinide (maybe tetravalent) elements as in the case of Th6Mn23-type compounds.   Table 4 (21-47 for R-containing compounds, and 48-51 for U-or Th-containing compounds).

Boron-Added Pd0.75Mn0.25 Alloy
The disordered face-centered-cubic (fcc) Pd1−xMnx alloys are the well studied spin-glass system [46]. The crystal structure is described by the space group of Fm3 m (No. 225), possessing only the 4a site (see Figure 5), that is randomly occupied by Pd and Mn atoms. At x = 0.25, the spin-glass  Table 4 (21-47 for R-containing compounds, and 48-51 for U-or Th-containing compounds).

Boron-Added Pd 0.75 Mn 0.25 Alloy
The disordered face-centered-cubic (fcc) Pd 1−x Mn x alloys are the well studied spin-glass system [46]. The crystal structure is described by the space group of Fm3m (No. 225), possessing only the 4a site (see Figure 5), that is randomly occupied by Pd and Mn atoms. At x = 0.25, the spin-glass transition temperature T SG is 45 K [47]. The Pd 0.75 Mn 0.25 alloy can incorporate boron atoms [48,49] with the solubility limit of approximately x~0.16 in Pd 0.75 Mn 0.25 B x . According to the structure report [50] of PdH x with the same structure, interstitial atoms prefer the octahedral sites (the 4b site, see Figure 5). The volume expansion occurs with the increase of x as shown in Table 2. Figure 6a shows the isothermal magnetization curves of Pd 0.75 Mn 0.25 B x at room temperature, which demonstrate the emergence of room temperature ferromagnetism by the slight addition of boron atoms. To elucidate the impact of the boron addition on the effective magnetic moment of Mn atoms, temperature dependences of inverse dc magnetization 1/χ dc are measured as shown in Figure 6b. All measured samples follow the Curie-Weiss law above T C (see the solid lines in Figure 6b, and the extracted effective magnetic moment µ eff and the Weiss temperature θ are summarized in Table 2). µ eff of the parent compound is 4.85 µ B /Mn, which is once reduced by the boron addition, that is a signature of hybridization between the Mn and boron atoms. However, the value grows with increasing T C up to 390 K, which is the maximum value in this system. Furthermore, µ eff is comparable to the saturation moment in Figure 6a. These facts suggest that the hybridization would be rather weak due to the wide space of the octahedral cavity and/or the rather low occupancy derived from being born at the octahedral site (e.g., 12.5% at x = 0.125). In the latter case, a Mn atom near the boron atom possesses a reduced magnetic moment and that with no neighboring boron atom would show an enhanced moment. Above x = 0.148, both T C and µ eff are reduced, which designates the dominating orbital hybridization between Mn and boron atoms, which is also consistent with the small change in the unit cell volume.      The magnetic phase diagram is constructed as shown in Figure 7a,b. At lower x, a coexistence of the fcc phase and the ordered derivative phase (AuCu3) of fcc occurs [48,49]. The latter phase is The magnetic phase diagram is constructed as shown in Figure 7a,b. At lower x, a coexistence of the fcc phase and the ordered derivative phase (AuCu 3 ) of fcc occurs [48,49]. The latter phase is responsible for another magnetic ordering at T M [49]. The spin-glass state survives under the emergence of FM state, and both ordering temperatures seem to compete with each other. Therefore, this system is considered to be the typical Mn-based compound presenting the additional formation of the FM state, coexisting with the spin-glass state of Mn atoms at low temperatures. Figure 7c displays the relationship between the T C and the unit cell volume. The vertical blue line is drawn at the volume of the parent compound showing only the spin-glass state. A linear correlation between the T C and the unit cell volume in the region of V = 60-64 Å 3 indicates a T C higher than 300 K even at the blue line, which means a possible abrupt birth of FM exchange coupling. responsible for another magnetic ordering at TM [49]. The spin-glass state survives under the emergence of FM state, and both ordering temperatures seem to compete with each other. Therefore, this system is considered to be the typical Mn-based compound presenting the additional formation of the FM state, coexisting with the spin-glass state of Mn atoms at low temperatures. Figure 7c displays the relationship between the TC and the unit cell volume. The vertical blue line is drawn at the volume of the parent compound showing only the spin-glass state. A linear correlation between the TC and the unit cell volume in the region of V = 60-64 Å 3 indicates a TC higher than 300 K even at the blue line, which means a possible abrupt birth of FM exchange coupling. We remarked on the transport properties of Pd0.75Mn0.25Bx. Figure 8 shows the temperature dependences of electrical resistivity, which highlight the rather large temperature dependence below TC even in the disordered alloy. Considering that the spin-glass parent compound shows a weak temperature dependence, the FM interaction might produce some coherence effect on the electrical conductivity. We remarked on the transport properties of Pd 0.75 Mn 0.25 B x . Figure 8 shows the temperature dependences of electrical resistivity, which highlight the rather large temperature dependence below T C even in the disordered alloy. Considering that the spin-glass parent compound shows a weak temperature dependence, the FM interaction might produce some coherence effect on the electrical conductivity. We remarked on the transport properties of Pd0.75Mn0.25Bx. Figure 8 shows the temperature dependences of electrical resistivity, which highlight the rather large temperature dependence below TC even in the disordered alloy. Considering that the spin-glass parent compound shows a weak temperature dependence, the FM interaction might produce some coherence effect on the electrical conductivity.  3.6. Boron-Added Sm 2 Mn 8 Al 9 Sm 2 Mn 8 Al 9 is isostructural to R 2 Fe 17 N x , as mentioned in Section 2.1, and allows the interstitial B atoms at the 9e site (see Figure 9) [51]. In the present case, Mn and Al atoms randomly occupy the Zn sites of hexagonal Th 2 Zn 17 -type structure (space group: R3m, No. 166). There is only one Wyckoff position 6c for Sm atoms, but Mn and Al atoms have four Wyckoff positions 6c, 9d, 18f and 18h, which are tentatively represented by Mn1, Mn2, Mn3 and Mn4, respectively, in Figure 9. The boron concentration dependences of lattice parameters and unit cell volume for Sm 2 Mn 8 Al 9 B x , determined with the help of the Rietveld refinement program [52], are listed in Table 3. The solubility limit would be x~1. ∆V/V at x ≥ 0.1 is much smaller than that of Pd 0.75 Mn 0.25 B x , which suggests the stronger orbital hybridization between Mn and boron atoms in Sm 2 Mn 8 Al 9 B x (see also Section 2.4). We note here that the nearest neighbor atoms of boron are Mn3 and Mn4, which amount to 71% of all Mn atoms (see also Figure 9). This may lead to a large increase in hybridization under a small variation of volume by the interstitial atoms.

Boron-Added Sm2Mn8Al9
Sm2Mn8Al9 is isostructural to R2Fe17Nx, as mentioned in Section 2.1, and allows the interstitial B atoms at the 9e site (see Figure 9) [51]. In the present case, Mn and Al atoms randomly occupy the Zn sites of hexagonal Th2Zn17-type structure (space group: R3 m, No. 166). There is only one Wyckoff position 6c for Sm atoms, but Mn and Al atoms have four Wyckoff positions 6c, 9d, 18f and 18h, which are tentatively represented by Mn1, Mn2, Mn3 and Mn4, respectively, in Figure 9. The boron concentration dependences of lattice parameters and unit cell volume for Sm2Mn8Al9Bx, determined with the help of the Rietveld refinement program [52], are listed in Table 3. The solubility limit would be x ~ 1. ∆V/V at x ≥ 0.1 is much smaller than that of Pd0.75Mn0.25Bx, which suggests the stronger orbital hybridization between Mn and boron atoms in Sm2Mn8Al9Bx (see also Section 2.4). We note here that the nearest neighbor atoms of boron are Mn3 and Mn4, which amount to 71% of all Mn atoms (see also Figure 9). This may lead to a large increase in hybridization under a small variation of volume by the interstitial atoms.   The isothermal magnetization curves are measured at room temperature as shown in Figure 10a, which have revealed the abrupt emergence of ferromagnetism at even a small amount of boron atoms. The temperature dependences of 1/χ dc demonstrate that boron-added samples follow the Curie-Weiss law above T C (see the solid lines in Figure 10b). Sm 2 Mn 8 Al 9 shows a FM behavior below approximately 100 K. Taking into account that the La-counterpart does not show a magnetic ordering at that temperature (see the ac magnetization χ ac results in the inset of Figure 10b), the low-temperature magnetic transition in the parent compound is due to Sm ions. The µ eff and Weiss temperature of each sample are summarized in Table 3. As x is increased, µ eff is rapidly reduced, which is indicative of strong orbital hybridization, which is also supported by the weak x dependence of the unit cell volume. It is to be pointed out that the paramagnetic Mn moment abruptly forms the FM state at room temperature even when the density of boron atoms is substantially low. Under a strong orbital hybridization between Mn and boron atoms, the lattice expansion might not be essential for the formation of FM coupling at room temperature.  The isothermal magnetization curves are measured at room temperature as shown in Figure 10a, which have revealed the abrupt emergence of ferromagnetism at even a small amount of boron atoms. The temperature dependences of 1/χdc demonstrate that boron-added samples follow the Curie-Weiss law above TC (see the solid lines in Figure 10b). Sm2Mn8Al9 shows a FM behavior below approximately 100 K. Taking into account that the La-counterpart does not show a magnetic ordering at that temperature (see the ac magnetization χac results in the inset of Figure 10b), the lowtemperature magnetic transition in the parent compound is due to Sm ions. The µeff and Weiss temperature of each sample are summarized in Table 3. As x is increased, µeff is rapidly reduced, which is indicative of strong orbital hybridization, which is also supported by the weak x dependence of the unit cell volume. It is to be pointed out that the paramagnetic Mn moment abruptly forms the FM state at room temperature even when the density of boron atoms is substantially low. Under a strong orbital hybridization between Mn and boron atoms, the lattice expansion might not be essential for the formation of FM coupling at room temperature. The magnetic properties of Sm2Mn8Al9Bx are summarized as a magnetic phase diagram in Figure  11a. The χac (T) measurements indicate the survival of magnetic ordering due to Sm ions at approximately 85 K, which is independent of the room temperature FM state and seems to disappear at x = 1. The high-temperature TC due to Mn atoms shows a shallow minimum at x = 0.75. The inset of Figure 11a is the high-temperature TC vs. V plot, in which V at x = 0 is denoted by the vertical blue The magnetic properties of Sm 2 Mn 8 Al 9 B x are summarized as a magnetic phase diagram in Figure 11a. The χ ac (T) measurements indicate the survival of magnetic ordering due to Sm ions at approximately 85 K, which is independent of the room temperature FM state and seems to disappear at x = 1. The high-temperature T C due to Mn atoms shows a shallow minimum at x = 0.75. The inset of Figure 11a is the high-temperature T C vs. V plot, in which V at x = 0 is denoted by the vertical blue line as in Figure 7c. The observation of room temperature ferromagnetism at both sides of the vertical line strongly suggests that the birth of room temperature FM coupling between Mn atoms is not related to the volume change. The effect of volume change, as studied well in rare-earth Fe-based magnets, is triggered by the appearance of the FM state due to Mn atoms. 12 magnets, is triggered by the appearance of the FM state due to Mn atoms.
The temperature dependence of ρ exhibits a drastic change as shown in Figure 11b. The carriers would show a localized nature up to x = 0.5, however, metallic behavior is observed above x = 0.75. The localized behavior is also reported in isostructural Mn compounds [53] such as Gd2MnxAl17−x and Tb2MnxAl17−x. The shift to the itinerant Mn moment with increasing x and/or some coherence effect of the FM Mn-moment observed in Pd0.75Mn0.25Bx would be responsible for the metallic temperature dependence.

Brief Summary of Mn-Based Compounds
As in the rare-earth Fe-based compounds, the interstitial atoms give rise to the enhancement of FM interaction in the weak hybridization regime leading to the appearance of room temperature ferromagnetism. However, the Mn compounds surveyed above manifest the change or additional formation of magnetism by the interstitial atoms, while many rare-earth Fe-based parent compounds are already ferromagnets. The change from paramagnetic to FM state is observed in hydrogenabsorbed Th6Mn23, hydrogen-absorbed YMn2 or Sm2Mn8Al9Bx. The result of Mn5Si3Cx thin film may be a rare example of change from the AFM to FM state by the interstitial atoms. In Pd0.75Mn0.25Bx, the room temperature ferromagnetism is induced by a slight addition of boron, while the lowtemperature magnetic ground state of the parent compound is unchanged. This can be regarded as an example of the additional formation of magnetism by interstitial atoms. It should be noted that, in some cases, the change or additional formation of a magnetic state seems to abruptly occur, which is valuable for future research. We note here that the magnetic structures have been divided into FM and AFM, although some compounds may show a more complicated state such as canted AFM, and spiral AFM. In the future, discussion taking into account a more microscopic mechanism of the magnetic ordering would be necessary.
We add the comment on Mn-based Heusler compounds, which show a rich variety of physical properties such as the topological Hall effect, shape-memory and so on [54][55][56]. While many Mnbased Heusler compounds show ferromagnetism, a Heusler compound, allowing interstitial atoms, has not been reported to our knowledge. The temperature dependence of ρ exhibits a drastic change as shown in Figure 11b. The carriers would show a localized nature up to x = 0.5, however, metallic behavior is observed above x = 0.75. The localized behavior is also reported in isostructural Mn compounds [53] such as Gd 2 Mn x Al 17−x and Tb 2 Mn x Al 17−x . The shift to the itinerant Mn moment with increasing x and/or some coherence effect of the FM Mn-moment observed in Pd 0.75 Mn 0.25 B x would be responsible for the metallic temperature dependence.

Brief Summary of Mn-Based Compounds
As in the rare-earth Fe-based compounds, the interstitial atoms give rise to the enhancement of FM interaction in the weak hybridization regime leading to the appearance of room temperature ferromagnetism. However, the Mn compounds surveyed above manifest the change or additional formation of magnetism by the interstitial atoms, while many rare-earth Fe-based parent compounds are already ferromagnets. The change from paramagnetic to FM state is observed in hydrogen-absorbed Th 6 Mn 23 , hydrogen-absorbed YMn 2 or Sm 2 Mn 8 Al 9 B x . The result of Mn 5 Si 3 C x thin film may be a rare example of change from the AFM to FM state by the interstitial atoms. In Pd 0.75 Mn 0.25 B x , the room temperature ferromagnetism is induced by a slight addition of boron, while the low-temperature magnetic ground state of the parent compound is unchanged. This can be regarded as an example of the additional formation of magnetism by interstitial atoms. It should be noted that, in some cases, the change or additional formation of a magnetic state seems to abruptly occur, which is valuable for future research. We note here that the magnetic structures have been divided into FM and AFM, although some compounds may show a more complicated state such as canted AFM, and spiral AFM. In the future, discussion taking into account a more microscopic mechanism of the magnetic ordering would be necessary.
We add the comment on Mn-based Heusler compounds, which show a rich variety of physical properties such as the topological Hall effect, shape-memory and so on [54][55][56]. While many Mn-based Heusler compounds show ferromagnetism, a Heusler compound, allowing interstitial atoms, has not been reported to our knowledge.

Candidate Showing a Change in Magnetism by Interstitial Atoms
Mn compounds may be a superior platform to examine the change in the magnetic state or the additional formation of new magnetic coupling by interstitial atoms. We noted that, in some cases, the change to FM state or the additional formation of the FM state would occur irrespectively of interstitial atom-induced volume change. Notwithstanding, hereafter, we continue to discuss, based on the unit cell volume, because the results of Mn 5 Si 3 C x , Pd 0.75 Mn 0.25 B x and Sm 2 Mn 8 Al 9 B x are not well analyzed systematically by comparing several compounds with the same crystal structure. In other words, the magnetic ordering temperature vs. unit cell volume plot as shown in Figure 2 could not be constructed due to the absence of well investigated other Mn compounds in these crystal structures. Therefore, at the present stage, we believe that it is still valuable to survey the magnetic structures of compounds using the unit cell volumes. In this subsection, especially bearing a change in magnetism by interstitial atoms in mind, we seek a qualifying Mn compound using the magnetic ordering temperature vs. a unit cell volume plot. We selected the crystal structures allowing the site occupation of interstitial atoms: Ni 3 Sn, ThMn 12 , Fe 2 P, Ni 2 In, LiAlSi, TiNiSi, AuCu and Cu 2 Sb-type structures.

Ni 3 Sn-Type Structure
This hexagonal structure (P6 3 /mmc, No. 194) is recently attractive as a topological AFM substance [20] represented by Mn 3 Sn. All Ni 3 Sn-type compounds displayed in Figure 12a possess only one crystallographic 6h site for the Mn atom. As seen in Figure 12a, only the AFM state has been observed and a change to FM state would be difficult. compounds with VEC = 7.7 showing the AFM state, all compounds undergo FM as one. In FM compounds, TC tends to increase as the VEC is increased. If interstitial atoms change VEC from 7.33 to 7.7, a change between FM and AFM states might be possible, which may be a new phenomenon induced by interstitial atoms.

TiNiSi-Type Structure
This is the famous orthorhombic structure with the space group Pnma (No. 62). Mn atoms occupy the 4c site. As shown in Figure 13b, the border between AFM and FM states may be situated at V = 180-200 Å 3 . Moreover, only the compounds accompanied by VEC = 5.3 show the FM ground state. The interstitial alloying with the change in VEC might be effective for a magnetic state change.

AuCu-Type Structure
The AuCu-type structure is tetragonal with the space group of P4/mmm (No. 123). Except for MnRh2Sb, Mn atoms occupy two crystallographic sites called the 1a and 1c sites. In MnRh2Sb, Mn sites are reduced to the 1a one. As in the LiAlSi-type structure, the magnetic ordering type would be correlated with VEC; the AFM or paramagnetic state is observed for a VEC larger than 8, and the FM state for VEC = 5 or 7.5. Under a fixed VEC value, TN systematically decreases with increasing volume, while the volume dependence of TC is very weak for ferromagnets.

Cu2Sb-Type Structure
This is the tetragonal structure (P4/nmm, No. 129), in which Mn atoms occupy the 2a site. Contrary to the prediction of the Bethe-Slater curve, a volume expansion favors an AFM state. The magnitude of the magnetic ordering temperature is likely suppressed by expanding the volume.   Table 4.

ThMn 12 -Type Structure
The tetragonal ThMn 12 -type structure (I4/mmm, No. 139) is well studied in rare-earth Fe-based compounds (see Section 2.2). In the Mn compounds with this structure, Mn atoms occupy the 8f, 8i, 8j sites. As shown in Figure 12b, the magnetic properties are dominated by the AFM state. We note that, except for compounds No. 55-60, the AFM orderings of compounds containing rare-earth elements are triggered by the magnetic moments of rare-earth, and Mn atoms do not carry moments. Basically, with increasing volume, T N tends to be suppressed. In this class of compounds, a change from AFM to a paramagnetic state of Mn atoms by interstitial atoms may be anticipated.

Fe 2 P-Type Structure
The compounds with the hexagonal Fe 2 P-type structure (P62m, No. 189) displayed in Figure 12c possess the 3g site for the Mn atom, except Mn 2 P and Mn 2 As, in which there exists the 3g and 3f sites for Mn atoms. Figure 12c indicates a possible crossover from paramagnetic to FM state across the volume of approximately 115-120 Å 3 for compounds with only the 3g Mn site.

Ni 2 In-Type Structure
The Ni 2 In-type structure is hexagonal with the space group of P6 3 /mmc (No. 194). Each compound has only one Mn site of the 2a. In this structure, T N at a smaller volume tends to be suppressed as the volume is expanded and seems to transform to T C with further increasing volume (Figure 12d). The compounds with the ThMn 12 -, the Fe 2 P-or the Ni 2 In-type structure may be good candidates for examining a change in magnetic state by interstitial atoms as a function of the unit cell volume.

LiAlSi-Type Structure
This structure is cubic with the space group of F43m (No. 216). Each compound in Figure 13a possesses one Mn site with the 4a, 4b or 4c, depending on the literature. The magnetic ordering types of the cubic compounds would be classified by the VEC, which is different from the results of Figure 12a-d. The VEC values are denoted under the chemical formulae in Figure 13a. Except for the compounds with VEC = 7.7 showing the AFM state, all compounds undergo FM as one. In FM compounds, T C tends to increase as the VEC is increased. If interstitial atoms change VEC from 7.33 to 7.7, a change between FM and AFM states might be possible, which may be a new phenomenon induced by interstitial atoms. Table 4. Structural and magnetic properties of Mn-based compounds. T order is the magnetic ordering temperature. FM, AFM and PM mean ferromagnetic, antiferromagnetic and paramagnetic, respectively. Data are from the references listed in the table.

TiNiSi-Type Structure
This is the famous orthorhombic structure with the space group Pnma (No. 62). Mn atoms occupy the 4c site. As shown in Figure 13b, the border between AFM and FM states may be situated at V = 180-200 Å 3 . Moreover, only the compounds accompanied by VEC = 5.3 show the FM ground state. The interstitial alloying with the change in VEC might be effective for a magnetic state change.

AuCu-Type Structure
The AuCu-type structure is tetragonal with the space group of P4/mmm (No. 123). Except for MnRh 2 Sb, Mn atoms occupy two crystallographic sites called the 1a and 1c sites. In MnRh 2 Sb, Mn sites are reduced to the 1a one. As in the LiAlSi-type structure, the magnetic ordering type would be correlated with VEC; the AFM or paramagnetic state is observed for a VEC larger than 8, and the FM state for VEC = 5 or 7.5. Under a fixed VEC value, T N systematically decreases with increasing volume, while the volume dependence of T C is very weak for ferromagnets.

Cu 2 Sb-Type Structure
This is the tetragonal structure (P4/nmm, No. 129), in which Mn atoms occupy the 2a site. Contrary to the prediction of the Bethe-Slater curve, a volume expansion favors an AFM state. The magnitude of the magnetic ordering temperature is likely suppressed by expanding the volume.

R 5 Si 3 B x
The Mn 5 Si 3 -type R 5 Si 3 allows for interstitial boron atoms, which leads to volume reduction with an increasing boron concentration [118]. For the parent compound with x = 0, the AFM orderings are observed in R = Gd and Tb at T N = 75 and 69 K, respectively. On the other hand, R = Dy and Ho show the FM orderings at T C = 120 and 11 K, respectively. By adding boron atoms, the T N of R = Gd and Tb are slightly reduced to 67 K in both cases, and the changes of T C in R = Dy and Ho are also subtle; T C remains at 120 K in R = Dy and is slightly enhanced to 15 K in R = Ho. In each compound, µ eff does not so largely depend on the boron addition, which means a weak hybridization between rare earth and boron atoms.

NdScSiC x
This compound crystallizes into the tetragonal La 2 Sb-type structure (I4/mmm, No. 139) [119]. The interstitial carbon atoms expand the a axis and contract the c axis. The latter fact, in particular, enhances the chemical bonding between Nd and C and decreases T C = 171 K in the parent NdScSi to 50 K at x = 0.5. Taking into account that the 4f orbital of a rare-earth atom is usually well localized, the drastic change in magnetic ordering temperature is unexpected as in R 5 Si 3 B x . Thus, the large modification of T C in this compound is very interesting.

Application of Mn-Based Magnetic Materials
The change in magnetism between FM and AFM states or the additional formation of new magnetic coupling at rather high temperatures by interstitial atoms is substantially valuable in a magnetic device integrated with both FM and AFM materials due to the easy on-demand control of magnetism in the fabrication process. At the present stage, Mn-based compounds fulfill the requirement of change or the additional formation of magnetism, while in Fe-based compounds, only improvements of FM properties are extensively investigated and the change (or the additional formation) of the magnetic state is not well explored. Focusing on the research area of the permanent magnet, a rare-earth Mn-based permanent magnet is still missing, although the MnBi-type magnets are well known. Based on the Bethe-Slater curve, Mn atoms favor the FM state with expanding Mn-Mn distance. Therefore, the density of Mn could not be so increased as in rare-earth Fe-based permanent magnets, resulting in a smaller saturation magnetization. However, there exists a large gap of BH energy product between the NdFeB magnets and ferrite magnets, and a rare-earth Mn-based permanent magnet may be a good candidate filling the gap [120].

Towards Further on-Demand Control of Magnetism
Further improvement of magnetic properties in the on-demand control would be achieved by another strategy such as a composition effect and carrier doping. If a metallurgical phase diagram of a target compound possesses a homogeneity range, the magnetic ordering temperature often varies with the atomic composition. For example, the T C of Tb 2 Co 2 Ga ranges rather widely from 75 to 145 K by changing the starting composition [121]. Such a composition effect is reported in other compounds [122][123][124] such as Nd 3 Pd 20 Ge 6 , Tb 3 Co 3 Ga and Mn 1+x Ga. In many cases, the crystal structure parameters slightly change, which heavily affects the magnetic exchange interactions.
The magnetic anisotropy energy is one of the important factors in characterizing a ferromagnet. It is known that it can be tuned by doping, mainly due to the variation of the density of states near the Fermi level. The doping effect is reported in, for example, Ni 2 MnGa, SmCo 5-x Fe, MnBi, Nd 2 Fe 17 X (X = C or N), Ce 2 AuP 3 and so on [125][126][127][128][129]. Taking into account the crystal symmetry, which is related to the magnetic anisotropy energy, a lower crystal symmetry with more tunable crystal parameters might be favorable.

Comments on Control of Magnetism by External Field
Another interesting control of magnetism is the manipulation of spin by external fields such as the electric field and the optical light. For example, the employment of an electric double-layer transistor has achieved a control of magnetism by weak voltage [130]. Recently, an optical change in magnetism through the Kondo effect has been reported [131]. The interstitial atoms may precisely control the magnetism so that only a small magnitude of external field is required for the device working, and the power consumption can be highly reduced.

Comments on Critical Behavior
From the fundamental viewpoint, research into critical behavior is interesting. Actually, in strongly correlated electron systems, there have been plenty of studies for seeking a quantum critical point under the suppression of magnetism [132][133][134][135][136]. We speculate that a formation of FM exchange coupling above room temperature would be a discontinuous phenomenon as mentioned in the results of Mn-based compounds. While it is not well investigated for rare-earth Fe-based compounds, we note that RFe 11 TiC x and RFe 11 TiC x show a finite change in T C at an infinitely zero value of volume expansion depending on R species [137].

Summary
This review surveyed the studies of interstitial atoms in rare-earth Fe-, Mn-, rare-earth-based magnetic materials, especially focusing on the Mn-based compounds since the effect of interstitial atoms is not well investigated. The light elements would occupy the interstitial sites in the respective crystal structure, and change the unit cell volume, although the degree of change depends on the strength of orbital hybridization between the magnetic and interstitial atoms. The light elements are occasionally essential for the stabilization of the desired crystal structure. In the abundant studies of rare-earth Fe-based permanent magnets, the role of interstitial atoms seems to be restricted to the enhancements of T C and saturated magnetization, and the change in easy magnetization direction. In several Mn-based compounds, magnetic ordering temperature and the magnetization also tends to be increased with the increasing density of interstitial atoms after the appearance of FM states. However, it is peculiar for the Mn-based compounds that the change or additional formation of magnetism by interstitial atoms is possible: the change from the AFM (paramagnetic) to FM state or the additional formation of the FM state coexisting with the ground state of Mn atoms in the parent compound. Furthermore, the FM exchange coupling would abruptly emerge under a slight addition of interstitial elements, and this is an important research topic for a deep understanding of interstitial atom engineering. The candidates of Mn-based compounds, possibly showing the change in magnetism by interstitial atoms, are briefly discussed. We note that not only the unit cell volume but also the VEC should be taken into account to design the change in magnetism. The change between AFM and FM states by just controlling the number of interstitial atoms is a very promising elemental technology for making highly functional magnetic devices integrated with both FM and AFM materials without a large lattice mismatch.