⇦Chinese Physics Letters, 2023, Vol. 40, No. 6, Article code 067502Review Possible Room-Temperature Ferromagnetic Semiconductors Jing-Yang You (尤景阳)1*, Xue-Juan Dong (董雪娟)2, Bo Gu (顾波)3*, and Gang Su (苏刚)3* Affiliations 1Department of Physics, Faculty of Science, National University of Singapore, 117551, Singapore 2Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 3Kavli Institute for Theoretical Sciences, and CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China Received 11 April 2023; accepted manuscript online 15 May 2023; published online 29 May 2023 *Corresponding authors. Email: phyjyy@nus.edu.sg; gubo@ucas.ac.cn; gsu@ucas.ac.cn Citation Text: You J Y, Dong X J, Gu B et al. 2023 Chin. Phys. Lett. 40 067502    Abstract Magnetic semiconductors integrate the dual characteristics of magnets and semiconductors. It is difficult to manufacture magnetic semiconductors that function at room temperature. Here, we review a series of our recent theoretical predictions on room-temperature ferromagnetic semiconductors. Since the creation of two-dimensional (2D) magnetic semiconductors in 2017, there have been numerous developments in both experimental and theoretical investigations. By density functional theory calculations and model analysis, we recently predicted several 2D room-temperature magnetic semiconductors, including CrGeSe$_3$ with strain, CrGeTe$_3$/PtSe$_2$ heterostructure, and technetium-based semiconductors (TcSiTe$_3$, TcGeSe$_3$, and TcGeTe$_3$), as well as PdBr$_3$ and PtBr$_3$ with a potential room-temperature quantum anomalous Hall effect. Our findings demonstrated that the Curie temperature of these 2D ferromagnetic semiconductors can be dramatically enhanced by some external fields, such as strain, construction of heterostructure, and electric field. In addition, we proposed appropriate doping conditions for diluted magnetic semiconductors, and predicted the Cr doped GaSb and InSb as possible room-temperature magnetic semiconductors.
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 DOI:10.1088/0256-307X/40/6/067502 © 2023 Chinese Physics Society Article Text Charge and spin are two inherent properties of electron. By changing the electron charge, conventional electronic devices serve a multitude of purposes. The size of the device has decreased due to continual performance improvement and is now getting close to the atomic scale's physical limit. Research of spintronics, which controls the electron spin, has gradually advanced to outperform conventional electronic devices, leading to a number of novel physical phenomena, including the tunnel magnetoresistance, anomalous hall effect, and topological insulator.[1-3] However, many new physical phenomena have not yet found practical applications. One major limitation is that some behavior can only be detected in the laboratory at low temperatures, whereas practical applications require them to be realized at room temperature. The foundational material for creating spintronic devices is magnetic semiconductor, an important spintronic material that combines two crucial properties of magnetism and semiconductor and has undergone extensive studies.[4] By incorporating magnetism into semiconductors, researchers can achieve faster data storage, sensors, capacitors, logic devices, etc.[5,6] using the two degrees of freedom of charge and spin of electrons. Spintronics ushers in the next generation of semiconducting devices for spin control.[7-9] Diluted magnetic semiconductors (DMSs) are made by doping ordinary semiconductors with magnetic elements to achieve spin controlled electrical and optical properties.[10,11] (Ga,Mn)As is the most widely and deeply explored magnetic semiconductor,[7] whose current Curie temperature $T_{\rm c}$ record is around 200 K[12] and is still much below room temperature. Fabricating two-dimensional (2D) van der Waals (vdW) magnetic semiconductors is another path to high-temperature magnetic semiconductors.[13] Ising-type ferromagnetism with a $T_{\rm c}$ of 45 K was discovered experimentally in single-layer CrI$_3$ in 2017.[14] Another experimental group obtained a 2D magnetic semiconductor bilayer CrGeTe$_3$ with a $T_{\rm c}$ of 28 K in the same year.[15] For practical usage, the ferromagnetic (FM) $T_{\rm c}$ of magnetic semiconductors is required to be above room temperature. Developing magnetic semiconductors that operate at room temperature was designated one of 125 unsolved scientific mysteries listed in the journal $Science$ in 2005.[16] As a result, finding more possible room-temperature FM semiconductors is essential, and developing mechanisms of increase in $T_{\rm c}$ is also indispensable. Possible Room-Temperature Ferromagnetic Semiconductors: Cr-doped GaSb and InSb. In the past decades, DMSs have been the major interest of magnetic semiconductor study.[17-20] The p-type semiconductor (Ga,Mn)As is a representative material, and its current maximum Curie temperature $T_{\rm c}$ is about 200 K. The FM Li(Zn,Mn)As was predicted in 2007.[21] Then, p-type FM semiconductor Li(Zn,Mn)As was developed.[22] The p-type Li(Zn,Mn)P with $T_{\rm c}$ of 34 K[22] and Li(Cd,Mn)P with $T_{\rm c}$ of 45 K[23] were experimentally realized. Experimentally, Mn-doped p-type FM semiconductor (Ba,K)(Zn,Mn)$_2$As$_2$ with $T_{\rm c}$ as high as 230 K was discovered in 2013.[24,25] Since 2016, $T_{\rm c}$ above room temperature was achieved in iron-doped p-type FM (Ga,Fe)Sb.[26,27] Besides p-type DMSs, significant advancements have been made in the study of n-type DMSs. In 2017, the experiment was reported such that the $T_{\rm c}$ in iron-doped n-type FM semiconductor (In,Fe)Sb[28,29] was higher than room temperature. In 2019, an experiment obtained n-type Ba(Zn,Co)$_2$As$_2$ with $T_{\rm c}$ of 45 K.[30] Low $T_{\rm c}$'s were revealed in p-type (Ga,Mn)Sb[31] and (In,Mn)Sb[32] compared to high $T_{\rm c}$ obtained in p-type (Ga,Fe)Sb and n-type (In,Fe)Sb. Additionally, $T_{\rm c}$'s of 70 K and 90 K were exposed in Fe-doped n-type (In,Fe)As[33] and Mn-doped p-type (In,Mn)As,[34] respectively. Recently, significant progress has been made in vdW DMSs. V-doped/alloyed WSe$_2$ monolayer films[35] and semiconducting V-doped WS$_2$ monolayers[36] were reported to exhibit tunable ferromagnetism at room temperature. It was reported that vdW semiconductor Cr$_x$Ga$_{1-x}$Te crystals with bandgap of 1.62–1.66 eV displayed highly controllable above-room-temperature ferromagnetism.[37] Cr-doped 2H-MoTe$_2$ FM semiconductor with a notable $T_{\rm c}$ up to 275 K and strong out-of-plane magnetization was also observed.[38] Table 1 provides a summary of these findings, where we only list the properties at the optimal doping concentration for vdW DMSs. It is noted that the carrier type of semiconductor has an influence on the introduction of ferromagnetism. For the DMS with a large band gap, the impurity bound state (IBS) is generally located near the valence band (VB) maximum due to strong mixing between the impurity and VB, while typically no IBS appears below the conduction band (CB) minimum as a result of weak mixing between the impurity and CB.[39] Thus, p-type semiconductors are easier to achieve room-temperature ferromagnetism. However, for the DMS with a small band gap, the IBS sits within the band gap and FM coupling can be realized for both n-type and p-type carriers by picking appropriate hosts and impurities. In this case, the strength of ferromagnetism mainly depends on the doping concentration and chemical potential dependent magnetic correlation between impurities.
Table 1. The experimental $T_{\rm c}$, impurity concentration, carrier type, and band gap for several DMSs. Reproduced with permission.[40] Copyright 2020, American Physical Society.
Ferromagnetic Curie temperature Impurity Carrier types Band gap Reference
semiconductors $T_{\rm c}$ (K) concentration $N_{\rm m}$  (eV) (year)
(Ga,Fe)Sb 340 25% p-type 0.75 Ref. [26] (2016)
(Ga,Mn)Sb 25 2.3% p-type 0.75 Ref. [31] (2000)
(Ga,Mn)As 200 16% p-type 1.4 Ref. [12] (2011)
(In,Fe)Sb 385 35% n-type 0.17 Ref. [29] (2019)
(In,Mn)Sb 10 4% p-type 0.17 Ref. [32] (2008)
(In,Fe)As 70 8% n-type 0.42 Ref. [33] (2012)
(In,Mn)As 90 10% p-type 0.42 Ref. [34] (2006)
(Ba,K)(Zn,Mn)$_2$As$_2$ 230 15% p-type 0.2 Ref. [25] (2014)
Ba(Zn,Co)$_2$As$_2$ 45 4% n-type 0.2 Ref. [30] (2019)
Li(Zn,Mn)As 50 10% p-type 1.6 Ref. [22] (2011)
Li(Zn,Mn)P 34 10% p-type 2.0 Ref. [23] (2013)
Li(Cd,Mn)P 45 10% p-type 1.3 Ref. [41] (2019)
V-doped WSe$_2$ 300 4% Ref. [35] (2020)
V-doped WS$_2$ $> 300$ 2% p-type Ref. [36] (2020)
Cr-doped GaTe 314.9–329 2.4% 1.62–1.66 Ref. [37] (2021)
Cr-doped MoTe$_2$ 275 2.1% Ref. [38] (2021)
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Fig. 1. (a) The supercell with two unitcells of InSb and Fe doping concentration of 12.5%, and its (b) [100] and (c) [001] views, where only Fe atoms in the supercell are indicated. Reproduced with permission.[40] Copyright 2020, American Physical Society.
To comprehend the micromechanism of low $T_{\rm c}$ in (In,Fe)As, (Ga,Mn)Sb, (In,Mn)Sb, and (In,Mn)As, and of high $T_{\rm c}$ in Fe-doped GaSb and InSb, a comprehensive theoretical investigation has been carried out.[40] We investigate the impurity concentration ($N_{\rm m}$) dependent $T_{\rm c}$ for n-type (In,Fe)Sb using the Weiss formula $T_{\rm c}=\frac{2}{3k_{\scriptscriptstyle{\rm B}}}S(S+1)\sum_i z_iJ_i$, where $z_i$ is the coordination number and $J_i$ represents the exchange coupling of the $i$th nearest neighbor (NN). Only four kinds of NN hoppings with $R_1 = (0.5, 0, 0.5)a$, $R_2 = (1, 0, 0)a$, $R_3 = (1, 0.5, 0.5)a$ and $R_4 = (1, 1, 0)a$ are taken into account. For ease of simulation, we consider three possible $N_{\rm m}$ corresponding to one, two, and three Fe atom impurities in an InSb supercell (two unit cells) with In atoms: 12.5%, 25%, and 37.5%. We first study the 12.5% doping concentration as shown in Fig. 1(a), where the Fe impurities constitute a 2D square lattice with a lattice constant $a$ ($R_2$) and $z_i = \{0, 4, 0, 4\}$ may be seen from Figs. 1(b) and 1(c). As a result, using the Weiss formula in conjunction with the estimated exchange integrals, we can determine that the $T_{\rm c}$ for 12.5% n-type Fe-doped InSb is 120 K. In the case of 37.5% doping concentration, only four of total 56 doping configurations are not equivalent. The likelihoods of the four doping techniques are 2/7, 2/7, 1/7, and 2/7, respectively. These result in a respective $T_{\rm c}$ of 399.3, 375.7, 375.7, and 363.7 K. Thus, the average $T_{\rm c}$ for 37.5% n-type (In,Fe)Sb is calculated to be 379 K, which is close to the observed 385 K in Table 1 for 35% $N_{\rm m}$. Our calculations using the same methodology suggest that the three FM semiconductors Cr-doped InSb, InAs, and GaSb could have high $T_{\rm c}$ as listed in Table 2. CrGeSe$_3$ with Strain. An efficient method to modify the characteristics of 2D materials is strain engineering.[42] Applying a predetermined strain could cause the Dirac cones at $K$ and $K^{\prime}$ in graphene to shift in opposite directions, producing a pseudomagnetic field.[43] The optical band gap of monolayer MoS$_2$ diminishes linearly with uniaxial tensile strain. One of the practical ways to introduce strain is to attach 2D material to a deforming substrate.[44]
Table 2. $T_{\rm c}$ corresponding to distinct doping concentrations for several p-type DMSs. Reproduced with permission.[40] Copyright 2020, American Physical Society.
Doping concentration 12.5% 25% 37.5%
$T_{\rm c}$ (K) (In,Cr)Sb 62 214 406
(In,Cr)As 17 129 264
(Ga,Cr)Sb 242 534 851
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Fig. 2. Crystal structure of CrGeSe$_3$: (a) top and (b) side views. (c) Normalized magnetization of CrGeTe$_3$ and CrGeSe$_3$ under different strains. Reproduced with permission.[45] Copyright 2019, American Physical Society.
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Fig. 3. Enhanced magnetic properties of CrGeTe$_3$ by constructing heterostructures. Crystal structure of CrGeTe$_2$/PtSe$_2$ heterostructures of [(a), (b)] stack 1 and [(c), (d)] stack 2. (e) Calculated normalized Curie temperature and (f) magnetic anisotropic energy (MAE) of CrGeTe$_3$/PtSe$_2$ heterostructure. Reproduced with permission.[46] Copyright 2020, American Physical Society.
From an application standpoint, 2D semiconductors with high $T_{\rm c}$'s ($T_{\rm c} > 300$ K) are quite useful. Experiments have established that bilayer CrGeTe$_3$ and monolayer CrI$_3$ are FM with low transition temperatures.[14,15] There has been a lot of research on raising the $T_{\rm c}$ above 300 K as a result of the realization of stable 2D ferromagnetism in these materials. We suggested CrGeSe$_3$ as one of the promising FM semiconductors based on density functional theory (DFT) calculations by replacing Te in CrGeTe$_3$ with Se.[45] Figures 2(a) and 2(b) depict the crystal structure. Our calculations show that 2D CrGeSe$_3$ possesses an FM ground state with out-of-plane magnetization. Normalized magnetization curves of CrGeSe$_3$ and CrGeSe$_3$ are plotted in Fig. 2(c). The $T_{\rm c}$ of monolayer CrGeTe$_3$ is calculated to be 30 K, while the $T_{\rm c}$ of CrGeSe$_3$ is calculated to be 144 K, which could be improved to 326 K with a 3% tensile strain. Our calculations confirm that tensile strain is a successful way to raise the $T_{\rm c}$ for CrGeSe$_3$. CrGeTe$_3$/PtSe$_2$ Heterostructure. Thermal fluctuations generally make it difficult to maintain long-range isotropic ferromagnetism in 2D systems.[47] By increasing the magnetic anisotropy for a 2D system, stable ferromagnetism at finite temperature can be achieved. The magnetic anisotropy of 2D ferromagnets CrI$_3$ and CrGeTe$_3$ was thought of as the consequence of the synergy between the Kitaev interaction and single ion anisotropy (SIA),[48] where spin-orbit coupling (SOC) is crucial in determining the magnetic anisotropy. Additionally, when combined to form heterostructures, the physical characteristics of 2D vdW systems can be incredibly diverse. Due to the influence of different stacking structures on magnetism,[49,50] when monolayer PtSe$_2$ was attached to monolayer CrGeTe$_3$, we consider two configurations of the heterostructure, coined as stack 1 and stack 2, as illustrated in Figs. 3(a)–3(d),[46] where the attached PtSe$_2$ layer is positioned beneath the CrGeTe$_3$. The lattice constant $a$ expands to 7.22 Å for CrGeTe$_3$/PtSe$_2$ heterostructures, which is larger than that for monolayer CrGeTe$_3$ and can be treated as a tensile strain of 3.7%. $T_{\rm c}$ and MAE are investigated by DFT calculations for monolayer CrGeTe$_3$ and CrGeTe$_3$ with 3.7% tensile strain and heterostructures of stack 1 and stack 2. The temperature-dependent magnetic moment is plotted in Fig. 3(e). We find that the $T_{\rm c}$ for monolayer CrGeTe$_3$ with relaxed lattice parameters is 200 K, it increases to 464 K with solely 3.7% strain and increases to 693 and 625 K for stack 1 and stack 2, respectively. The MAE for heterostructures stacks 1 and 2 is likewise with percentages of 70% and 40% higher than that for the strained CrGeTe$_3$ as seen in Fig. 3(f). This indicates different mechanisms between attaching a layer PtSe$_2$ and implementing strain. Our results confirm that a proper non-magnetic layer can significantly enhance the ferromagnetism of CrGeTe$_3$ by constructing heterostructure. TcSiTe$_3$, TcGeSe$_3$, and TcGeTe$_3$. Large magnetic anisotropy is necessary to maintain the 2D magnetism at finite temperature.[47] Magneto-optic Kerr effect (MOKE) describes a basic magneto-optic phenomenon where plane polarized light reflected from the magnetic material changes into elliptical polarized light and rotates the polarized light plane, which is closely related to SOC. The quantum confinement effect,[51] Kerr rotation oscillation with the magnetosphere thickness,[52] and the significant correlation between MOKE and magnetic anisotropy[53] are just a few of the excitation phenomena connected to MOKE that have been discovered. Researchers have been looking for magnetic materials with large Kerr angles since MOKE has promising applications in magneto-optical memory devices.
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Fig. 4. (a) Crystal structure. (b) Temperature-dependent magnetization. Partial density of states (PDOS) of (c) TcSiTe$_3$ and (d) CrGeTe$_3$ monolayers. (e) Anomalous Hall conductivity (AHC) $\sigma_{xy}$. (f) Photo-energy dependent Kerr angle $\theta_{_{\scriptstyle \rm Kerr}}$ with experimental (open squares) and estimated (dotted line) values for bulk Fe included for comparison. Reproduced with permission.[54] Copyright 2020, American Physical Society.
Three stable 2D technetium (Tc) based FM semiconductors TcSiTe$_3$, TcGeSe$_3$, and TcGeTe$_3$ are proposed in this subsection[54] with the prototypical structure of CrGeTe$_3$[15] as shown in Fig. 4(a). The Monte Carlo (MC) simulation shows that the $T_{\rm c}$'s of monolayers TcSiTe$_3$, TcGeSe$_3$, and TcGeTe$_3$ are 538, 212, and 187 K respectively, which are far higher than that of CrGeTe$_3$ as presented in Fig. 4(b). Our results unveil that the spin magnetic moment $S$ of each Tc atom in these materials is about $2\mu_{\scriptscriptstyle{\rm B}}$, and the orbital magnetic moment is about $0.5\mu_{\scriptscriptstyle{\rm B}}$. The large orbital magnetic moment comes from the partially occupied $d$ orbital as seen in Fig. 4(c), which is the result of the comparable electron correlation effect and crystal field. On the contrary, for Cr$^{3+}$ in monolayer CrGeTe$_3$, the integer occupied $d$ orbital with three spin-up electrons occupying $t_{\rm 2g}$ and one spin-up electron occupying $e_{\rm g}$ results in $S=4 \mu_{\scriptscriptstyle{\rm B}}$ and quenched orbital moment as illustrated in Fig. 4(d). The enhanced SOC caused by a larger orbital magnetic moment leads to the large MAE, resulting in the Ising behavior of out-of-plane magnetization in these Tc-based materials. In addition, p-type and n-type TcGeTe$_3$ have large AHCs of $7.5 \times 10^{2}$ and $1.1 \times 10^{3}$ $\Omega\cdot$cm$^{-1}$ as given in Fig. 4(e), and its Kerr rotation angle is about 3.6$^{\circ}$, which is far greater than 0.8$^{\circ}$ in metal Fe as shown in Fig. 4(f). Our findings suggest a series of high-temperature magnetic semiconductor materials with large SOC, which are of great significance for development of next-generation microelectronic spintronics devices. PdBr$_3$ and PtBr$_3$. The interplay of strong SOC and FM order can produce many topological effects,[55] and in a 2D system it may generate the quantum anomalous Hall effect (QAHE), where the edge state carries the quantized Hall conductance.[56-59] QAHE has promising prospective applications in low-consumption spintronic devices due to its non-dissipative chiral edge state.[60] However, in the present experiment, QAHE can only be accomplished at extremely low temperatures. Extremely low temperatures are typically used in experiments to suppress magnetic disorder so as to obtain complete quantization of QAHE in magnetically doped topological insulators.[61-65] To counteract the effects of magnetic disorder, 2D intrinsic FM semiconductors carrying nonzero Chern numbers $C$ are excellent candidates to achieve QAHE.[66-72] Honeycomb lattice, which was inspired by Haldane's pioneering work,[56] is widely regarded as a suitable platform for QAHE.[73-76]
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Fig. 5. (a) Crystal structure of PdBr$_3$ (PtBr$_3$). (b) Temperature-dependent magnetization. The $C$ of several bands close to the Fermi level for (c) PdBr$_3$ and (d) PtBr$_3$ with SOC, respectively. AHC $\sigma_{xy}$ of (e) PdBr$_3$ and (f) PtBr$_3$. Edge states of (g) PdBr$_3$ and (h) PtBr$_3$. Reproduced with permission.[77] Copyright 2019, American Physical Society.
In this subsection, we introduce two monolayers PdBr$_3$ and PtBr$_3$ with honeycomb lattice[77] as plotted in Fig. 5(a), which has the prototype structure of CrI$_3$. We first validate the structural stability by phonon spectrum, formation energy, and free energy. The spin polarized calculations reveal that these two materials have FM ground state with out-of-plane magnetization and a much high MAE. We determine, using the four-state method, that the Kitaev anisotropy and SIA of two monolayers are both negative,[48] exhibiting the Ising behavior. As a result, an effective Hamiltonian can be written as $H_{\rm spin}=-\sum_{\langle i, j \rangle}JS_i^zS_j^z$, where $J$ stands for the NN exchange integral, $S$ for the spin operator, and $S_{i/j}^z$ for the spin operator. The $J$'s of PdBr$_3$ and PtBr$_3$ monolayers are estimated to be 79.2 meV and 84.9 meV, respectively. Their $T_{\rm c}$'s are determined by MC simulation[78] based on the above Ising model to be 350 K and 375 K [see Fig. 5(b)], respectively, demonstrating that PdBr$_3$ and PtBr$_3$ monolayers are possible room-temperature ferromagnets. The SOC opens a band gap $E_{\rm g}$ of 58.7 and 28.1 meV in monolayers PdBr$_3$ and PtBr$_3$, respectively, as presented in Figs. 5(c) and 5(d), and the $E_{\rm g}/K_{\scriptscriptstyle{\rm B}}$ value is greater than the room temperature. Since the generalized gradient approximation method typically underestimates the $E_{\rm g}$, the HSE06 approach is also utilized to obtain more precise $E_{\rm g}$, which gives the $E_{\rm g}$'s of 100.8 and 45 meV for PdBr$_3$ and PtBr$_3$, respectively. To explore the topological properties, the $C$ is calculated to be 1. As expected by the nonzero $C$, the AHC shows a plateau of $\sigma_{xy}=Ce^2/h=e^2/h$ as displayed in Figs. 5(e) and 5(f). The nonzero $C$ is directly connected to the number of non-trivial chiral edge states that appear in the energy gap, according to the principle of correspondence between bulk and edge states.[79] A chiral edge state that connects the CB and VB can be observed in Figs. 5(g) and 5(h). The aforementioned findings demonstrate that these two systems are capable of realizing the intrinsic room-temperature QAHE. It is noted that, in our recent theoretical study for the multi-orbital Hubbard models, the electron correlations can enhance the effective SOC by a factor of $1/[1-(2U^\prime-U-J_{\rm H})\rho_0]$.[80] Possible Mechanisms to Enhance the Curie Temperature—Appropriate Doping Conditions in DMSs. The chemical potential $\mu$ of n-type carriers is around 0.17 eV for Fe-doped InSb, which is towards the bottom of the CB. As seen in Figs. 6(a) and 6(b), the FM coupling between Fe impurities is found at $\mu = 0.17$ eV. Figures 6(c) and 6(d) illustrate the distance $R_{12}$-dependent $\langle M_{1\xi}^zM_{2\xi}^z \rangle$, where long-range FM couplings up to about 8 Å (1.4$a_0$) and 13 Å (2$a_0$) are observed for the $yz$ orbital and $xz$ orbital for the p-type and n-type cases, respectively, as plotted in Figs. 6(c) and 6(d). The FM coupling in the n-type case is stronger than that in the p-type case, as seen in comparison of Figs. 6(c) and 6(d), which is in line with the high $T_{\rm c}$ reported in the n-type case (In,Fe)Sb.[28,29] Moreover, our calculation in Fig. 6(c) indicates that high $T_{\rm c}$ in (In,Fe)Sb with p-type carriers is also achievable. For Fe-doped GaSb, $\mu$-dependent $\langle n_\xi \rangle$ and $\langle M_{1\xi}^zM_{2\xi}^z \rangle$ are given in Figs. 6(e) and 6(f). The FM couplings among $xy$, $xz$, and $yz$ orbitals are seen for the p-type carriers, while no FM coupling is found for the n-type carriers. This is in line with the high $T_{\rm c}$ recorded in the p-type (Ga,Fe)Sb.[26,27] Therefore, a shallow energy level and suitable chemical potential are crucial for obtaining high-$T_{\rm c}$ DMSs. In addition, our analysis shows that in the diluted impurity limit, the magnetic correlation $\langle M_{1}M_{2} \rangle$ of the Fe, Mn, and Cr impurities in the same semiconductor is comparable as listed in Table 3. Through comparing the results in Table 1, the high (low) $T_{\rm c}$ in these experiments is believed to be primarily caused by the high (low) impurity concentration. Our findings demonstrate that the absence of carriers introduced by Fe$^{3+}$ rather than a carrier-induced process is what causes high $T_{\rm c}$ in (Ga,Fe)Sb and (In,Fe)Sb. As a result, elevating the concentration of impurities while preserving the integrity of the crystal structure is the main strategy for raising $T_{\rm c}$ in DMSs, for instance, by selecting main semiconductors and suitable impurities to prevent the valence discrepancy during main doping.
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Fig. 6. For Fe doped InSb, $\mu$-dependent (a) occupation number $\langle n_\xi \rangle$ and (b) magnetic correlation $\langle M_{1\xi}^zM_{2\xi}^z \rangle$ with fixed first NN $R_{12}$, and $R_{12}$-dependent $\langle M_{1\xi}^zM_{2\xi}^z \rangle$ (c) for the p-type case with $\mu = 0$ eV and (d) for the n-type case with $\mu = 0.2$ eV. [(e), (f)] Similar to (a) and (b) for Fe doped GaSb. Reproduced with permission.[40] Copyright 2020, American Physical Society.
Table 3. The DFT + QMC results of the magnitude of $\langle M_{1}M_{2} \rangle$. QMC: quantum Monte Carlo. Reproduced with permission.[40] Copyright 2020, American Physical Society.
The maximum $\langle M_{1}M_{2} \rangle$ at the first NN Host semiconductors
InSb InAs GaSb
(gap = 0.17 eV) (gap = 0.42 eV) (gap = 0.75 eV)
Fe 0.07 (p-type) 0.14 (p-type) 0.1 (p-type)
0.09 (n-type)
Mn 0.14 (p-type) 0.13 (p-type) 0.13 (p-type)
0.07 (n-type)
Cr 0.13 (p-type) 0.12 (p-type) 0.12 (p-type)
0.08 (n-type)
Table 4. DFT results for Cr–Cr interaction of CrGeSe$_3$ in pristine and with 5% tensile strain. Reproduced with permission.[45] Copyright 2019, American Physical Society.
Strain (%) Hopping-matrix element $|V|$ (eV) Energy difference $|E_{\rm d}-E_{\rm p}|$ (eV)
$p_z-d_{z^2}$ $p_z-d_{xz}$ $p_z-d_{yz}$ $p_z-d_{x^2+y^2}$ $p_z-d_{xy}$ $p_z-d_{z^2}$ $p_z-d_{xz}$ $p_z-d_{yz}$ $p_z-d_{x^2+y^2}$ $p_z-d_{xy}$
0 0.25798 0.29527 0.38028 0.19004 0.44591 1.41503 0.44580 0.45512 0.58167 0.58117
5 0.25829 0.22747 0.30687 0.20152 0.41632 1.55224 0.09143 0.09775 0.69506 0.69395
Strain Control. The improvement of $T_{\rm c}$ by imposing strain for monolayer CrGeSe$_3$ could be interpreted using the super exchange model. The interaction between adjacent Cr atoms is intermediated by Se atoms. The coupling between 3$d$ orbital electrons of Cr and 4$p$ orbital of Se reads $J=|V|^2/(|E_{\rm d}-E_{\rm p}|)$, where $|V|$ is the hopping matrix between Cr-3$d$ and Se-4$p$ orbitals and $|E_{\rm d} - E_{\rm p}|$ is the corresponding energy difference. Calculated parameters for monolayer CrGeSe$_3$ under 0% and 5% tensile strain are displayed in Table 4. It demonstrates that applying strain barely affects the hopping matrix element. However, the energy difference between Cr-$p_z$ and Cr-3$d_{xz}/3d_{yz}$ is reduced to one fifth with 5% strain. As a result, the FM interaction between Cr atoms is strengthened, and the $T_{\rm c}$ rises above room temperature. In a recent paper, the magnetic force microscopy signal of room-temperature ferromagnetism was observed in highly strained nanoscale wrinkles in 2D CrGeTe$_3$, and the DFT calculations suggested that the room-temperature FM order is achievable in 2D CrGeTe$_3$ at 6–8% strain.[81] Construction of Heterostructure. Both $T_{\rm c}$ and MAE are elevated in heterostructures made of CrGeTe$_3$ and PtSe$_2$ compared to pristine CrGeTe$_3$. The mechanism of strain enhanced Curie temperature is analogous to that in CrGeSe$_3$. To comprehend the magnetic anisotropy enhancement in heterostructure CrGeTe$_3$/PtSe$_2$, a general Hamiltonian can be formulated as \begin{align} H_{\rm spin}=\sum_{\langle i, j \rangle, \alpha, \beta}J_{\alpha\beta}S_{i\alpha}S_{j\beta}+\sum_{i, \alpha, \beta}A_{\alpha\beta}S_{i\alpha}S_{i\beta}, \tag {1} \end{align} where $\alpha$ and $\beta$ represent the $x$, $y$, $z$ direction indices, $J_{\alpha\beta}$ is the exchange couplings, and $A_{\alpha\beta}$ is the SIA. The diagonal and off-diagonal elements of $J$ are presented in Tables 5 and 6. As shown in Table 5, all systems considered have FM interaction as evidenced by the negative value of the diagonal elements of $J_{\alpha\alpha}$. The off-diagonal elements $J_{\alpha\beta}$ given in Table 6 are much smaller than the diagonal ones, so the FM character is dominated by $J_{\alpha\alpha}$. For the system without inversion symmetry, the Dzyaloshinskii–Moriya (DM) interaction is expressed as $D_z=\frac{|J_{yz}-J_{xy}|}{2}$.[82] It is noted that for monolayer CrGeTe$_3$, without or with strain, $J_{\alpha\beta}$ equals $J_{\beta\alpha}$, resulting in vanished DM interaction. Noticeably, for CrGeTe$_3$/PtSe$_2$ heterostructures, finite values of $|J_{\alpha\beta}-J_{\beta\alpha}|$ imply the presence of DM interaction, which helps to enhance MAE.
Table 5. The diagonal elements of $J$ (in units of meV) for monolayer CrGeTe$_3$ and CrGeTe$_3$ with 3.7% tensile strain, and CrGeTe$_3$/PtSe$_2$ heterostructures stack 1 and stack 2. Reproduced with permission.[46] Copyright 2020, American Physical Society.
$J_{xx}$ $J_{yy}$ $J_{zz}$
CrGeTe$_3$-relaxation $-$8.068 $-$7.467 $-$8.316
CrGeTe$_3$-strain $-$11.314 $-$10.591 $-$11.620
Stack 1 $-$12.557 $-$11.813 $-$12.896
Stack 2 $-$12.290 $-$11.508 $-$12.623
Table 6. The off-diagonal elements. Reproduced with permission.[46] Copyright 2020, American Physical Society.
$J_{xy}$ $J_{yx}$ $J_{xz}$ $J_{zx}$ $J_{yz}$ $J_{zy}$ $|J_{xy}-J_{yz}|$ $|J_{xz}-J_{zx}|$ $|J_{yz}-J_{zy}|$
CrGeTe$_3$-relaxation $-$0.493 $-0.493$ 0.069 0.069 $-0.111$ $-0.111$ 0.000 0.000 0.000
CrGeTe$_3$-strain $-$0.593 $-0.593$ $-0.061$ $-0.061$ 0.102 0.102 0.000 0.000 0.000
Stack 1 $-$0.750 $-0.539$ $-0.053$ 0.228 $-0.188$ $-0.072$ 0.211 0.281 0.116
Stack 2 0.669 0.628 $-0.153$ 0.334 0.293 0.014 0.041 0.487 0.279
cpl-40-6-067502-fig7.png
Fig. 7. (a) Crystal structure of monolayer MnBi$_2$Te$_4$. (b) Direct band gap at the $\varGamma$ point, (c) MAE, (d) SOC strength, and (e) $T_{\rm c}$ versus electric field. (f) An illustration of the micromechanism causing increased $T_{\rm c}$ in an electric field. (g) Band structure and (h) AHC of MnBi$_2$Se$_2$Te$_2$. (i) Band structure, (j) AHC, and (k) edge states of MnBi$_2$S$_2$Te$_2$. Reproduced with permission.[83] Copyright 2021, American Physical Society.
Electric Field Control. There are many advantages in adjusting the properties of the energy band by introducing an electric field. For instance, the direction and size of an electric field can be easily modulated. It is simple to employ an electric field on materials in experiments.[84,85] Graphene is not easy to be applied to devices because of the zero band gap, which can be opened by applying a gate voltage.[86,87] Doping holes to partially occupy the flat band in a novel carbon monolayer can induce ferromagnetism.[88] The electric field can govern the transition between the direct and indirect band gaps in the InSe system.[89] The valley property of MoS$_2$ can be monitored by the electric field.[90,91] By introducing a built-in electric field through intercalation technology, an interlayer magnetic phase transition and strong magnetoelectric coupling can be induced.[92] The electric field can also adjust the channel resistance of a Cr$_2$Ge$_2$Te$_6$ thin layer[93] and the interlayer coupling of bilayer CrI$_3$[94,95] or be applied to a field-effect-transistor device,[96] and the combination of electric field and magnetism can realize the spin filter function.[97] An intrinsic electric polarization (electric field) induced by constructing Janus structure can tune the magnetic ground state and enhance SOC.[98] This subsection studies the regulation of an electric field on the characteristics of MnBi$_2$Te$_4$ monolayer.[83] The MnBi$_2$Te$_4$ monolayer is an intrinsic FM semiconductor with out-of-plane magnetization.[99] The band gap ($E_{\rm g}$) of MnBi$_2$Te$_4$ becomes narrower when the vertical electric field is applied, and the system enters the topological state with an electric field of 0.1 V/Å as shown in Fig. 7(b). In the range of electric field from 0 to 0.15 V/Å, the MAE rises from 0.1 to 5 meV and the $T_{\rm c}$ enhances from 13 to 61 K as given in Figs. 7(c) and 7(e). The promotion of SOC by the electric field is the primary cause of the rise in MAE [see Fig. 7(d)]. With the help of the super-exchange interaction model, it is found that the increase of $T_{\rm c}$ comes from the enhancement of the hybridization between 3$d$ electron of the Mn atom and 5$p$ electron of the Te atom and from the reduction of their energy difference as plotted in Fig. 7(f). Based on the above-mentioned findings, we find that an electric field can bring about a topological phase transition, strengthen SOC, and rise the $T_{\rm c}$. We suggest that the Janus structure MnBi$_2$Se$_2$Te$_2$ can realize the aforementioned topological phase transition. MnBi$_2$Se$_2$Te$_2$ is obtained by replacing the unit Te–Bi–Te in monolayer MnBi$_2$Te$_4$ with Se–Bi–Se. Owing to the different electronegativity of Te and Se atoms in monolayer MnBi$_2$Se$_2$Te$_2$, a self-generating polarization with an intensity of 0.44 $e$Å exists. The monolayer MnBi$_2$Se$_2$Te$_2$ is an intrinsic FM semiconductor with out-of-plane magnetization, and its $T_{\rm c}$ is 25 K. The energy gap of monolayer MnBi$_2$Se$_2$Te$_2$ without considering SOC is 0.5158 eV, which is much smaller than those of MnBi$_2$Te$_4$ (0.9088 eV) and MnBi$_2$Se$_4$ (0.8325 eV). After taking into account SOC, the band inversion occurs and $E_{\rm g}$ becomes 16 meV, resulting in the QAHE state with $C =1$ as shown in Figs. 7(g) and 7(h). A similar janus structure MnBi$_2$S$_2$Te$_2$, which has spontaneous polarization of 0.73 $e$Å, can be produced when Te is replaced with S, where S has stronger electronegativity. According to the above discussion, in monolayer MnBi$_2$S$_2$Te$_2$, there should be a larger $E_{\rm g}$ and a higher $T_{\rm c}$ than those in monolayer MnBi$_2$Se$_2$Te$_2$. The monolayer MnBi$_2$S$_2$Te$_2$ is also unveiled to be an intrinsic FM semiconductor with out-of-plane magnetization, $E_{\rm g}$ of 50 meV [see Fig. 7(i)] and $T_{\rm c}$ of 48 K. The $C$ of monolayer MnBi$_2$S$_2$Te$_2$ is 2, indicating a QAHE state with a quantized AHC plateau and two edge states in the energy gap, as shown in Figs. 7(j) and 7(k). This work will advance the study of novel electric field-controlled material properties and offer good materials for FM semiconductors with self-generating polarization to realize the QAHE. Discussion. Compared with metallic ferromagnets, it is challenging to make semiconductors with room-temperature long-range FM order. This is because for the room-temperature FM metals, such as Fe metal, the 3$d$ electrons contribute to the band structure with direct exchange interactions, while for the FM semiconductors, including DMSs, 2D CrI$_3$, and CrGeTe$_3$, their 3$d$ electrons are more localized, and the indirect exchange interactions are dominant, which appear to be much weaker than the direct exchange interactions. Carrier-mediated exchange interactions for DMSs have been extensively studied. Materials with very high doping concentrations are needed to produce room-temperature DMSs, such as the high-$T_{\rm c}$ DMSs in Table 1. In 2D vdW FM semiconductors, the super-exchange interactions have been demonstrated. For example, in 2D CrI$_3$ and CrGeTe$_3$, the indirect FM coupling strength is proportional to the hopping strength and inversely proportional to the energy level difference between 3$d$ orbitals of Cr and $p$ orbitals of I/Te. Therefore, to obtain room-temperature 2D FM semiconductors, materials with high hopping strength and small energy level difference are required, such as the high-$T_{\rm c}$ 2D FM semiconductors discussed above. In summary, we first go through five different types of high-temperature FM semiconductors with nice properties that are eagerly awaited for experimental confirmation. Then, we present four techniques for improving magnetism and $T_{\rm c}$ and explain their microscopic mechanisms, which can be applicable to currently well-studied/used magnetic semiconductors, and help to promote their performances. High-performance ferromagnet research is still in progress, but it also has various prospects and obstacles. Undoubtedly, advancements in this developing field will aid in the creation of fast non-volatile storage systems, where the magnetization direction not only governs charge transport but can also be electrically modified. Several groundbreaking investigations have been sparked by the integration of these ferromagnets with electronics and spintronics. However, to achieve this goal, some scientific and technological obstacles still need to be overcome, such as increasing magnetic doping concentration, manipulating and strengthening magnetic ordering, overcoming lattice mismatch, comprehending the FM exchange mechanism, creating new ferromagnets with high $T_{\rm c}$, and being aware of proximity effects. Finding experimentally synthesizable and air-stable FM materials is crucial for real-world device applications. Even though there have been many theoretical efforts devoted to 2D magnets with high $T_{\rm c}$, very little has been known about their mechanical and chemical stability. Exceptional 2D FM materials can now be designed on a new platform, thanks to the high-throughput and machine learning techniques in recent years, which will facilitate the discovery and implementation of high-temperature FM materials. Acknowledgments. This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 12074378 and 11834014), the Beijing Natural Science Foundation (Grant No. Z190011), the National Key R&D Program of China (Grant No. 2018YFA0305800), the Beijing Municipal Science and Technology Commission (Grant No. Z191100007219013), the Chinese Academy of Sciences (Grant Nos. YSBR-030 and Y929013EA2), and the Strategic Priority Research Program of Chinese Academy of Sciences (Grant Nos. XDB28000000 and XDB33000000). 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