Magnetism in two-dimensional materials⌗

In recent years a promising opportunity appeared that can push spintronic devices to a two-dimensional limit. Eliminating one dimension has the practical benefit of reducing the size and energy consumption of spintronic devices. Two-dimensional materials can in addition be tuned by gates and dopants. Two dimensional magnetism on the other hand has favorable properties such as larger magnetic domains, larger magnetic moment per atom and in general large directional magneto-crystalline anisotropy ( Citation: Sethulakshmi & al., 2019 Sethulakshmi, N., Mishra, A., Ajayan, P., Kawazoe, Y., Roy, A., Singh, A. & Tiwary, C. (2019). Magnetism in two-dimensional materials beyond graphene. Mater. Today, 27. 107–122. https://doi.org/10.1016/j.mattod.2019.03.015 ; Citation: Zhang & al., 2019 Zhang, W., Wong, P., Zhu, R. & Wee, A. (2019). Van der Waals magnets: Wonder building blocks for two-dimensional spintronics?. InfoMat, 1(4). 479–495. https://doi.org/10.1002/inf2.12048 ).

It is important to stress that this anisotropy is important not because it is beneficial for storing magnetic data, but because without it the magnetic order (ferromagnetic or antiferromagnetic) cannot exist. This is an important consequence of the Mermin-Wagner theorem ( Citation: Mermin & al., 1966 Mermin, N. & Wagner, H. (1966). Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic Heisenberg models. Phys. Rev. Lett., 17(22). 1133–1136. https://doi.org/10.1103/physrevlett.17.1133 ), that states that in a truly isotropic two-dimensional system a long-range magnetic order (ferromagnetic or antiferromagnetic) cannot exist at any finite temperature: any, whatever small, thermal excitation gives rise to large magnonic fluctuations strong enough to destroy any magnetic ordering. A magneto-crystalline anisotropy however circumvents this theorem as it opens up a magnonic excitation gap, giving rise to finite Curie/Néel temperatures below which a ferromagnetic/antiferromagnetic ordering exists.

The first two-dimensional crystal, graphene ( Citation: Katsnelson, 2007 Katsnelson, M. (2007). Graphene: Carbon in two dimensions. Mater. Today, 10(1-2). 20–27. https://doi.org/10.1016/s1369-7021(06)71788-6 ; Citation: Guinea & al., 2009 Guinea, F., Katsnelson, M. & Geim, A. (2009). Energy gaps and a zero-field quantum hall effect in graphene by strain engineering. Nat. Phys., 6(1). 30–33. https://doi.org/10.1038/nphys1420 ; Citation: Katsnelson & al., 2012 Katsnelson, M. & Fasolino, A. (2012). Graphene as a prototype crystalline membrane. Acc. Chem. Res., 46(1). 97–105. https://doi.org/10.1021/ar300117m ), was first discovered in 2004 ( Citation: Novoselov, 2004 Novoselov, K. (2004). Electric field effect in atomically thin carbon films. Science, 306(5696). 666–669. https://doi.org/10.1126/science.1102896 ; Citation: Novoselov & al., 2005 Novoselov, K., Geim, A., Morozov, S., Jiang, D., Katsnelson, M., Grigorieva, I., Dubonos, S. & Firsov, A. (2005). Two-dimensional gas of massless dirac fermions in graphene. Nature, 438(7065). 197–200. https://doi.org/10.1038/nature04233 ). A popular technique to produce a single layer of graphene is by exfoliating graphite. Here the different layers in graphite are held together by weak van-der-Waals forces. The same technique has led to the discovery of a multitude of new two-dimensional materials. Collectively, these materials are called van-der-Waals materials. These materials are often grouped in four distinct groups ( Citation: Sethulakshmi & al., 2019 Sethulakshmi, N., Mishra, A., Ajayan, P., Kawazoe, Y., Roy, A., Singh, A. & Tiwary, C. (2019). Magnetism in two-dimensional materials beyond graphene. Mater. Today, 27. 107–122. https://doi.org/10.1016/j.mattod.2019.03.015 ; Citation: Tsymbal & al., 2019 Tsymbal, E. & Žutić, I. (2019). Spintronics Handbook, Second Edition: Spin Transport and Magnetism: Volume Three: Nanoscale Spintronics and Applications. CRC Press. ; Citation: Liu & al., 2020 Liu, W., Bryan, M. & Xu, Y. (2020). Introduction to spintronics and 2D materials. InLiu, W. & Xu, Y. (Eds.), Spintronic 2D materials. (pp. 1–24). Elsevier. https://doi.org/10.1016/b978-0-08-102154-5.00001-1 ):

1. graphene based
2. 2D chalcogenides ( Citation: Yang & al., 2015 Yang, L., Sinitsyn, N., Chen, W., Yuan, J., Zhang, J., Lou, J. & Crooker, S. (2015). Long-lived nanosecond spin relaxation and spin coherence of electrons in monolayer MoS2 and WS2. Nat. Phys., 11(10). 830–834. https://doi.org/10.1038/nphys3419 ; Citation: Chhowalla & al., 2013 Chhowalla, M., Shin, H., Eda, G., Li, L., Loh, K. & Zhang, H. (2013). The chemistry of two-dimensional layered transition metal dichalcogenide nanosheets. Nature Chem, 5(4). 263–275. https://doi.org/10.1038/nchem.1589 ; Citation: Wang & al., 2012 Wang, Q., Kalantar-Zadeh, K., Kis, A., Coleman, J. & Strano, M. (2012). Electronics and optoelectronics of two-dimensional transition metal dichalcogenides. Nat. Nanotechnol., 7(11). 699–712. https://doi.org/10.1038/nnano.2012.193 ; Citation: Ugeda & al., 2015 Ugeda, M., Bradley, A., Zhang, Y., Onishi, S., Chen, Y., Ruan, W., Ojeda-Aristizabal, C., Ryu, H., Edmonds, M., Tsai, H., Riss, A., Mo, S., Lee, D., Zettl, A., Hussain, Z., Shen, Z. & Crommie, M. (2015). Characterization of collective ground states in single-layer NbSe2. Nat. Phys., 12(1). 92–97. https://doi.org/10.1038/nphys3527 ; Citation: Zhang & al., 2013 Zhang, Y., Chang, T., Zhou, B., Cui, Y., Yan, H., Liu, Z., Schmitt, F., Lee, J., Moore, R., Chen, Y., Lin, H., Jeng, H., Mo, S., Hussain, Z., Bansil, A. & Shen, Z. (2013). Direct observation of the transition from indirect to direct bandgap in atomically thin epitaxial MoSe2. Nat. Nanotechnol., 9(2). 111–115. https://doi.org/10.1038/nnano.2013.277 ; Citation: Ma & al., 2012 Ma, Y., Dai, Y., Guo, M., Niu, C., Zhu, Y. & Huang, B. (2012). Evidence of the existence of magnetism in pristine VX2 monolayers (X = s, se) and their strain-induced tunable magnetic properties. ACS Nano, 6(2). 1695–1701. https://doi.org/10.1021/nn204667z ; Citation: Butler & al., 2013 Butler, S., Hollen, S., Cao, L., Cui, Y., Gupta, J., Gutiérrez, H., Heinz, T., Hong, S., Huang, J., Ismach, A., Johnston-Halperin, E., Kuno, M., Plashnitsa, V., Robinson, R., Ruoff, R., Salahuddin, S., Shan, J., Shi, L., Spencer, M., Terrones, M., Windl, W. & Goldberger, J. (2013). Progress, challenges, and opportunities in two-dimensional materials beyond graphene. ACS Nano, 7(4). 2898–2926. https://doi.org/10.1021/nn400280c ; Citation: Zeng & al., 2012 Zeng, H., Dai, J., Yao, W., Xiao, D. & Cui, X. (2012). Valley polarization in MoS2 monolayers by optical pumping. Nat. Nanotechnol., 7(8). 490–493. https://doi.org/10.1038/nnano.2012.95 ; Citation: Xiao & al., 2012 Xiao, D., Liu, G., Feng, W., Xu, X. & Yao, W. (2012). Coupled spin and valley physics in monolayers ofMoS2and other group-VI dichalcogenides. Phys. Rev. Lett., 108(19). 196802. https://doi.org/10.1103/physrevlett.108.196802 ; Citation: Mak & al., 2010 Mak, K., Lee, C., Hone, J., Shan, J. & Heinz, T. (2010). Atomically ThinMoS2: A new direct-gap semiconductor. Phys. Rev. Lett., 105(13). 136805. https://doi.org/10.1103/physrevlett.105.136805 ; Citation: Radisavljevic & al., 2011 Radisavljevic, B., Radenovic, A., Brivio, J., Giacometti, V. & Kis, A. (2011). Single-layer MoS2 transistors. Nat. Nanotechnol., 6(3). 147–150. https://doi.org/10.1038/nnano.2010.279 )
3. 2D halides ( Citation: Leng & al., 2018 Leng, K., Abdelwahab, I., Verzhbitskiy, I., Telychko, M., Chu, L., Fu, W., Chi, X., Guo, N., Chen, Z., Chen, Z., Zhang, C., Xu, Q., Lu, J., Chhowalla, M., Eda, G. & Loh, K. (2018). Molecularly thin two-dimensional hybrid perovskites with tunable optoelectronic properties due to reversible surface relaxation. Nat. Mater., 17(10). 908–914. https://doi.org/10.1038/s41563-018-0164-8 ; Citation: Dou & al., 2015 Dou, L., Wong, A., Yu, Y., Lai, M., Kornienko, N., Eaton, S., Fu, A., Bischak, C., Ma, J., Ding, T., Ginsberg, N., Wang, L., Alivisatos, A. & Yang, P. (2015). Atomically thin two-dimensional organic-inorganic hybrid perovskites. Science, 349(6255). 1518–1521. https://doi.org/10.1126/science.aac7660 ; Citation: Sachs & al., 2013 Sachs, B., Wehling, T., Novoselov, K., Lichtenstein, A. & Katsnelson, M. (2013). Ferromagnetic two-dimensional crystals: Single layers of K2CuF4. Phys. Rev. B, 88(20). 201402. https://doi.org/10.1103/physrevb.88.201402 ; Citation: Zhang & al., 2019 Zhang, W., Li, Y., Jin, H. & She, Y. (2019). Two-dimensional transition-metal halide CoBr3 with spin-polarized dirac cone. Phys. Chem. Chem. Phys., 21(32). 17740–17745. https://doi.org/10.1039/c9cp03337h ; Citation: Spanopoulos & al., 2019 Spanopoulos, I., Hadar, I., Ke, W., Tu, Q., Chen, M., Tsai, H., He, Y., Shekhawat, G., Dravid, V., Wasielewski, M., Mohite, A., Stoumpos, C. & Kanatzidis, M. (2019). Uniaxial expansion of the 2D ruddlesden–popper perovskite family for improved environmental stability. J. Am. Chem. Soc., 141(13). 5518–5534. https://doi.org/10.1021/jacs.9b01327 )
4. 2D oxides ( Citation: Jia & al., 2019 Jia, Y., Zhao, M., Gou, G., Zeng, X. & Li, J. (2019). Niobium oxide dihalides NbOX2: A new family of two-dimensional van der Waals layered materials with intrinsic ferroelectricity and antiferroelectricity. Nanoscale Horiz., 4(5). 1113–1123. https://doi.org/10.1039/c9nh00208a ; Citation: Wang & al., 2018 Wang, J., Zhou, Y., He, M., Wangyang, P., Lu, Y. & Gu, L. (2018). Electrolytic approach towards the controllable synthesis of NiO nanocrystalline and self-assembly mechanism of Ni(OH)2 precursor under electric, temperature and magnetic fields. CrystEngComm, 20(17). 2384–2395. https://doi.org/10.1039/c8ce00263k ; Citation: Yao & al., 2020 Yao, W. & Li, Y. (2020). Ferrimagnetism and anisotropic phase tunability by magnetic fields in Na2Co2TeO6. Phys. Rev. B, 101(8). 085120. https://doi.org/10.1103/physrevb.101.085120 ; Citation: Yang & al., 2019 Yang, J., Zeng, Z., Kang, J., Betzler, S., Czarnik, C., Zhang, X., Ophus, C., Yu, C., Bustillo, K., Pan, M., Qiu, J., Wang, L. & Zheng, H. (2019). Formation of two-dimensional transition metal oxide nanosheets with nanoparticles as intermediates. Nat. Mater., 18(9). 970–976. https://doi.org/10.1038/s41563-019-0415-3 ; Citation: Kalantar-zadeh & al., 2016 Kalantar-zadeh, K., Ou, J., Daeneke, T., Mitchell, A., Sasaki, T. & Fuhrer, M. (2016). Two dimensional and layered transition metal oxides. Appl. Mater. Today, 5. 73–89. https://doi.org/10.1016/j.apmt.2016.09.012 ).

The first category consists of materials that are a derivative of graphene — such as fluorated graphene ( Citation: Feng & al., 2016 Feng, W., Long, P., Feng, Y. & Li, Y. (2016). Two-dimensional fluorinated graphene: Synthesis, structures, properties and applications. Adv. Sci., 3(7). 1500413. https://doi.org/10.1002/advs.201500413 ; Citation: Nair & al., 2010 Nair, R., Ren, W., Jalil, R., Riaz, I., Kravets, V., Britnell, L., Blake, P., Schedin, F., Mayorov, A., Yuan, S., Katsnelson, M., Cheng, H., Strupinski, W., Bulusheva, L., Okotrub, A., Grigorieva, I., Grigorenko, A., Novoselov, K. & Geim, A. (2010). Fluorographene: A two-dimensional counterpart of Teflon. Small, 6(24). 2877–2884. https://doi.org/10.1002/smll.201001555 ) — and materials that have a similar hexagonal crystal structure – such as hexagonal boron nitride ( Citation: Zhang & al., 2017 Zhang, K., Feng, Y., Wang, F., Yang, Z. & Wang, J. (2017). Two dimensional hexagonal boron nitride (2D-hBN): Synthesis, properties and applications. J. Mater. Chem. C, 5(46). 11992–12022. https://doi.org/10.1039/c7tc04300g ). The group of 2D chalcogenides consists of crystals containing at least one chalgenide atom (e.g. S, Se, Te). A commonly studied subgroup are the transition metal dichalcogenides (TMDs), whose crystal formula is given by MX$_2$. Here M is a transition metal (e.g. Mo, W) and X is a chalcogenide. The third group, 2D halides, contain crystals following a similar formula as the TMDs, e.g. MX$_2$ and MX$_3$, but with X being a halogen (e.g. Cl, Br, I). The magnetic moments in most of these crystals are strongly located on the metal atoms in a honeycomb array ( Citation: Chittari & al., 2020 Chittari, B., Lee, D., Banerjee, N., MacDonald, A., Hwang, E. & Jung, J. (2020). Carrier- and strain-tunable intrinsic magnetism in two-dimensional MAX3 transition metal chalcogenides. Phys. Rev. B, 101(8). 085415. https://doi.org/10.1103/physrevb.101.085415 ). The last group of 2D oxides ranges from simple crystals (e.g. ZnO) to more complex (e.g. Na$_2$Co$_2$TeO$_6$). Some examples of crystals in the last three groups are presented in Table 1.

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The groups of two-dimensional chalgonides and halides are currently a popular topic of investigation as they provide access to many physical properties not found in other two-dimensional materials. The electronic properties, e.g. bandgap, of these crystals are highly tunable to doping, strain, and chemical composition. Furthermore, a large variety of magnetic phases are also found among these materials (see Figure 1).

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Apart from the crystals in Table 1, there is also a large amount of two-dimensional crystals that are neither ferro- or antiferromagnetic. The possibility to induce (anti)ferromagnetism in such a crystal is attractive as it gives access to even a wider range of material parameters. In general one can undertake three ways:

1. doping and defects ( Citation: Gonzalez-Herrero & al., 2016 Gonzalez-Herrero, H., Gomez-Rodriguez, J., Mallet, P., Moaied, M., Palacios, J., Salgado, C., Ugeda, M., Veuillen, J., Yndurain, F. & Brihuega, I. (2016). Atomic-scale control of graphene magnetism by using hydrogen atoms. Science, 352(6284). 437–441. https://doi.org/10.1126/science.aad8038 ; Citation: Han, 2016 Han, W. (2016). Perspectives for spintronics in 2D materials. APL Mater., 4(3). 032401. https://doi.org/10.1063/1.4941712 ; Citation: Ugeda & al., 2010 Ugeda, M., Brihuega, I., Guinea, F. & Gómez-Rodríguez, J. (2010). Missing atom as a source of carbon magnetism. Phys. Rev. Lett., 104(9). 096804. https://doi.org/10.1103/physrevlett.104.096804 ; Citation: Han & al., 2014 Han, W., Kawakami, R., Gmitra, M. & Fabian, J. (2014). Graphene spintronics. Nat. Nanotechnol., 9(10). 794–807. https://doi.org/10.1038/nnano.2014.214 ; Citation: Boukhvalov & al., 2011 Boukhvalov, D. & Katsnelson, M. (2011). Sp-electron magnetic clusters with a large spin in graphene. ACS Nano, 5(4). 2440–2446. https://doi.org/10.1021/nn103510c ; Citation: Boukhvalov & al., 2008 Boukhvalov, D., Katsnelson, M. & Lichtenstein, A. (2008). Hydrogen on graphene: Electronic structure, total energy, structural distortions and magnetism from first-principles calculations. Phys. Rev. B, 77(3). 035427. https://doi.org/10.1103/physrevb.77.035427 ; Citation: Nair & al., 2013 Nair, R., Tsai, I., Sepioni, M., Lehtinen, O., Keinonen, J., Krasheninnikov, A., Castro Neto, A., Katsnelson, M., Geim, A. & Grigorieva, I. (2013). Dual origin of defect magnetism in graphene and its reversible switching by molecular doping. Nat Commun, 4(1). 2010. https://doi.org/10.1038/ncomms3010 );
2. magnetic proximity ( Citation: Gonzalez-Herrero & al., 2016 Gonzalez-Herrero, H., Gomez-Rodriguez, J., Mallet, P., Moaied, M., Palacios, J., Salgado, C., Ugeda, M., Veuillen, J., Yndurain, F. & Brihuega, I. (2016). Atomic-scale control of graphene magnetism by using hydrogen atoms. Science, 352(6284). 437–441. https://doi.org/10.1126/science.aad8038 ; Citation: Han, 2016 Han, W. (2016). Perspectives for spintronics in 2D materials. APL Mater., 4(3). 032401. https://doi.org/10.1063/1.4941712 ; Citation: Ugeda & al., 2010 Ugeda, M., Brihuega, I., Guinea, F. & Gómez-Rodríguez, J. (2010). Missing atom as a source of carbon magnetism. Phys. Rev. Lett., 104(9). 096804. https://doi.org/10.1103/physrevlett.104.096804 ; Citation: Han & al., 2014 Han, W., Kawakami, R., Gmitra, M. & Fabian, J. (2014). Graphene spintronics. Nat. Nanotechnol., 9(10). 794–807. https://doi.org/10.1038/nnano.2014.214 );
3. strain engineering ( Citation: Shen & al., 2016 Shen, T., Penumatcha, A. & Appenzeller, J. (2016). Strain engineering for transition metal dichalcogenides based field effect transistors. ACS Nano, 10(4). 4712–4718. https://doi.org/10.1021/acsnano.6b01149 ; Citation: Chittari & al., 2020 Chittari, B., Lee, D., Banerjee, N., MacDonald, A., Hwang, E. & Jung, J. (2020). Carrier- and strain-tunable intrinsic magnetism in two-dimensional MAX3 transition metal chalcogenides. Phys. Rev. B, 101(8). 085415. https://doi.org/10.1103/physrevb.101.085415 ).

The first is by chemically doping the crystal with $d$ or $f$ elements (e.g. Mn, Eu, Cr) or by introducing defects. For example, by doping GaAs with Mn atoms, exchange is introduced between local and delocalized spins giving rise to ferromagnetism ( Citation: Dietl & al., 2014 Dietl, T. & Ohno, H. (2014). Dilute ferromagnetic semiconductors: Physics and spintronic structures. Rev. Mod. Phys., 86(1). 187–251. https://doi.org/10.1103/revmodphys.86.187 ). Selectively introducing defects on only one sublattice in graphene can also induce weak ferromagnetism – a consequence of Lieb’s theorem ( Citation: Lieb, 1989 Lieb, E. (1989). Two theorems on the Hubbard model. Phys. Rev. Lett., 62(16). 1927–1927. https://doi.org/10.1103/physrevlett.62.1927.5 ; Citation: Lieb, 1989 Lieb, E. (1989). Two theorems on the Hubbard model. Phys. Rev. Lett., 62(10). 1201–1204. https://doi.org/10.1103/physrevlett.62.1201 ). The second approach is by bringing the crystal in vicinity with a (anti)ferromagnetic material. For example one can deposit graphene on top of an insulating ferromagnet such as YIG or EuS to create ferromagnetic graphene ( Citation: Wang & al., 2015 Wang, Z., Tang, C., Sachs, R., Barlas, Y. & Shi, J. (2015). Proximity-induced ferromagnetism in graphene revealed by the anomalous hall effect. Phys. Rev. Lett., 114(1). 016603. https://doi.org/10.1103/physrevlett.114.016603 ; Citation: Leutenantsmeyer & al., 2016 Leutenantsmeyer, J., Kaverzin, A., Wojtaszek, M. & Wees, B. (2016). Proximity induced room temperature ferromagnetism in graphene probed with spin currents. 2D Mater., 4(1). 014001. https://doi.org/10.1088/2053-1583/4/1/014001 ). The third approach makes use of the fact that electronic properties change by means of strain. A compressive strain of $\sim1\\%$ in FeSiS$_3$ is predicted ( Citation: Chittari & al., 2020 Chittari, B., Lee, D., Banerjee, N., MacDonald, A., Hwang, E. & Jung, J. (2020). Carrier- and strain-tunable intrinsic magnetism in two-dimensional MAX3 transition metal chalcogenides. Phys. Rev. B, 101(8). 085415. https://doi.org/10.1103/physrevb.101.085415 ) to transition it from a ferromagnetic ground state to a antiferromagnetic ground state.

Other properties such as spin-orbit coupling ( Citation: Soumyanarayanan & al., 2016 Soumyanarayanan, A., Reyren, N., Fert, A. & Panagopoulos, C. (2016). Emergent phenomena induced by spin–orbit coupling at surfaces and interfaces. Nature, 539(7630). 509–517. https://doi.org/10.1038/nature19820 ; Citation: Liu & al., 2011 Liu, L., Moriyama, T., Ralph, D. & Buhrman, R. (2011). Spin-torque ferromagnetic resonance induced by the spin hall effect. Phys. Rev. Lett., 106(3). 036601. https://doi.org/10.1103/physrevlett.106.036601 ) and magnetocrystalline anisotropy ( Citation: Soumyanarayanan & al., 2016 Soumyanarayanan, A., Reyren, N., Fert, A. & Panagopoulos, C. (2016). Emergent phenomena induced by spin–orbit coupling at surfaces and interfaces. Nature, 539(7630). 509–517. https://doi.org/10.1038/nature19820 ; Citation: Dieny & al., 2017 Dieny, B. & Chshiev, M. (2017). Perpendicular magnetic anisotropy at transition metal/oxide interfaces and applications. Rev. Mod. Phys., 89(2). 025008. https://doi.org/10.1103/revmodphys.89.025008 ) can be tuned or induced as well by placing a magnetic layer on top of a heavy metal layer such as Pt, Ta or W ( Citation: Hellman & al., 2017 Hellman, F., Hoffmann, A., Tserkovnyak, Y., Beach, G., Fullerton, E., Leighton, C., MacDonald, A., Ralph, D., Arena, D., Dürr, H., Fischer, P., Grollier, J., Heremans, J., Jungwirth, T., Kimel, A., Koopmans, B., Krivorotov, I., May, S., Petford-Long, A., Rondinelli, J., Samarth, N., Schuller, I., Slavin, A., Stiles, M., Tchernyshyov, O., Thiaville, A. & Zink, B. (2017). Interface-induced phenomena in magnetism. Rev. Mod. Phys., 89(2). 025006. https://doi.org/10.1103/revmodphys.89.025006 ). As an example, recently strong current induced spin-orbit torques were measured in a bilayer consisting of Fe$_3$GeTe$_2$ and Pt ( Citation: Wang & al., 2019 Wang, X., Tang, J., Xia, X., He, C., Zhang, J., Liu, Y., Wan, C., Fang, C., Guo, C., Yang, W., Guang, Y., Zhang, X., Xu, H., Wei, J., Liao, M., Lu, X., Feng, J., Li, X., Peng, Y., Wei, H., Yang, R., Shi, D., Zhang, X., Han, Z., Zhang, Z., Zhang, G., Yu, G. & Han, X. (2019). Current-driven magnetization switching in a van der Waals ferromagnet Fe3GeTe2. Sci. Adv., 5(8). eaaw8904. https://doi.org/10.1126/sciadv.aaw8904 ).

Dirac ferro- and antiferromagnets⌗

In my PhD Thesis I investigated the role of conducting electrons in assisting in the manipulation and relaxation of magnetic moments in ferro- and antiferromagnets. Specifically, we study two-dimensional ferro- and antiferromagnets where the conducting electrons have a linear energy dispersion. Such electrons are called Dirac fermions and the system as a whole is referred to as either a Dirac ferro- or antiferromagnet.

Dirac fermions were first found in graphene ( Citation: Novoselov, 2004 Novoselov, K. (2004). Electric field effect in atomically thin carbon films. Science, 306(5696). 666–669. https://doi.org/10.1126/science.1102896 ; Citation: Novoselov & al., 2005 Novoselov, K., Jiang, D., Schedin, F., Booth, T., Khotkevich, V., Morozov, S. & Geim, A. (2005). Two-dimensional atomic crystals. Proc. Natl. Acad. Sci. U.S.A., 102(30). 10451–10453. https://doi.org/10.1073/pnas.0502848102 ; Citation: Novoselov & al., 2005 Novoselov, K., Geim, A., Morozov, S., Jiang, D., Katsnelson, M., Grigorieva, I., Dubonos, S. & Firsov, A. (2005). Two-dimensional gas of massless dirac fermions in graphene. Nature, 438(7065). 197–200. https://doi.org/10.1038/nature04233 ) and soon after in topological insulators ( Citation: Hasan & al., 2010 Hasan, M. & Kane, C. (2010). Colloquium: Topological insulators. Rev. Mod. Phys., 82(4). 3045–3067. https://doi.org/10.1103/revmodphys.82.3045 ; Citation: König & al., 2008 König, M., Buhmann, H., W. Molenkamp, L., Hughes, T., Liu, C., Qi, X. & Zhang, S. (2008). The quantum spin hall effect: Theory and experiment. J. Phys. Soc. Jpn., 77(3). 031007. https://doi.org/10.1143/jpsj.77.031007 ; Citation: Qi & al., 2010 Qi, X. & Zhang, S. (2010). The quantum spin hall effect and topological insulators. Phys. Today, 63(1). 33–38. https://doi.org/10.1063/1.3293411 ; Citation: Cayssol, 2013 Cayssol, J. (2013). Introduction to dirac materials and topological insulators. C. R. Phys., 14(9-10). 760–778. https://doi.org/10.1016/j.crhy.2013.09.012 ). The Dirac ferromagnet that I studied in ( Citation: Sokolewicz & al., 2019 Sokolewicz, R., Ado, I., Katsnelson, M., Ostrovsky, P. & Titov, M. (2019). Spin-torque resonance due to diffusive dynamics at the surface of a topological insulator. Physical Review B, 99(21). 214444. Retrieved from https://journals.aps.org/prb/abstract/10.1103/PhysRevB.99.214444 ) is inspired by a bilayer consisting of a topological insulator and a ferromagnetic insulator. The model can be directly used to describe for example a bilayer consisting of Bi$_2$Te$_3$ or Bi$_2$Se$_3$ and YIG or EuS.

In my PhD work a particular antiferromagnet is studied on a honeycomb lattice. Although most of the antiferromagnets presented in Table 1 have a honeycomb lattice, the majority are unfortunately non-metallic, or are not in the Néel antiferromagnetic phase (illustrated in Figure 1b). Recent DFT calculations ( Citation: Chittari & al., 2020 Chittari, B., Lee, D., Banerjee, N., MacDonald, A., Hwang, E. & Jung, J. (2020). Carrier- and strain-tunable intrinsic magnetism in two-dimensional MAX3 transition metal chalcogenides. Phys. Rev. B, 101(8). 085415. https://doi.org/10.1103/physrevb.101.085415 ) predict however a metallic Néel antiferromagnetic phase for the following monolayer transition metal trichalgenides: FeSiSe$_3$, FeSiTe$_3$, VGeTe$_3$, MnGeS$_3$, FeGeSe$_3$, FeGeTe$_3$, NiGeSe$_3$, MnSnS$_3$, MnSnS$_3$, MnSnSe$_3$, FeSnSe$_3$, NiSnS$_3$. Experimental realizations of one of these materials could serve as a testing place for our model.

Furthermore, not every hexagonal metallic Néel antiferromagnet can accurately be described with a simple Dirac Hamiltonian. The closest experimentally realized material would be CuMnAs that has a square lattice, but can host Dirac electrons. The results from my PhD work ( Citation: Baglai & al., 2020 Baglai, M., Sokolewicz, R., Pervishko, A., Katsnelson, M., Eriksson, O., Yudin, D. & Titov, M. (2020). Giant anisotropy of gilbert damping in a rashba honeycomb antiferromagnet. Physical Review B, 101(10). 104403. Retrieved from https://journals.aps.org/prb/abstract/10.1103/PhysRevB.101.104403 ; Citation: Sokolewicz & al., 2019 Sokolewicz, R., Ghosh, S., Yudin, D., Manchon, A. & Titov, M. (2019). Spin-orbit torques in a rashba honeycomb antiferromagnet. Physical Review B, 100(21). 214403. Retrieved from https://journals.aps.org/prb/abstract/10.1103/PhysRevB.100.214403 ; Citation: Sokolewicz, 2020 Sokolewicz, R. (2020). Spintronics in two-dimensional conducting dirac ferro-and antiferromagnets  (PhD thesis). [Sl: sn] Retrieved from https://repository.ubn.ru.nl/bitstream/handle/2066/225934/225934.pdf ) can at least qualitatively describe the current induced phenomena found in CuMnAs.

Other Dirac antiferromagnets exist as well (e.g. TaCoTe$_2$ ( Citation: Wang, 2017 Wang, J. (2017). Antiferromagnetic dirac semimetals in two dimensions. Phys. Rev. B, 95(11). 115138. https://doi.org/10.1103/physrevb.95.115138 ), Zr$_2$Si ( Citation: Shao & al., 2018 Shao, X., Liu, X., Zhang, X., Wang, J. & Zhao, M. (2018). Zr2Si: An antiferromagnetic dirac MXene. Phys. Chem. Chem. Phys., 20(6). 3946–3952. https://doi.org/10.1039/c7cp08108a ) , BaFe$_2$As$_2$ and SrFe$_2$As$_2$ ( Citation: Chen & al., 2017 Chen, Z., Wang, L., Song, Y., Lu, X., Luo, H., Zhang, C., Dai, P., Yin, Z., Haule, K. & Kotliar, G. (2017). Two-dimensional massless dirac fermions in antiferromagnetic AFe2As2 (A=Ba,Sr). Phys. Rev. Lett., 119(9). 096401. https://doi.org/10.1103/physrevlett.119.096401 ), EuCd$_2$As$_2$ ( Citation: Ma & al., 2020 Ma, J., Wang, H., Nie, S., Yi, C., Xu, Y., Li, H., Jandke, J., Wulfhekel, W., Huang, Y., West, D., Richard, P., Chikina, A., Strocov, V., Mesot, J., Weng, H., Zhang, S., Shi, Y., Qian, T., Shi, M. & Ding, H. (2020). Emergence of nontrivial low-energy dirac fermions in antiferromagnetic EuCd 2 as 2. Adv. Mater., 32(14). 1907565. https://doi.org/10.1002/adma.201907565 ), MnBi$_2$Te$_4$ ( Citation: Swatek & al., 2020 Swatek, P., Wu, Y., Wang, L., Lee, K., Schrunk, B., Yan, J. & Kaminski, A. (2020). Gapless dirac surface states in the antiferromagnetic topological insulator MnBi2Te4. Phys. Rev. B, 101(16). 161109. https://doi.org/10.1103/physrevb.101.161109 )), but they suffer from symmetry protected gapless states and low exchange energies between localized and conducting electrons. As such, they are unideal when it comes to manipulating the antiferromagnetic order.

The ideal Dirac antiferromagnet would be an antiferromagnetic version of graphene. Though currently non-existent, it has recently been predicted that antiferromagnetism can be induced in graphene by bringing it in proximity with MnPSe$_3$ ( Citation: Högl & al., 2020 Högl, P., Frank, T., Zollner, K., Kochan, D., Gmitra, M. & Fabian, J. (2020). Quantum anomalous hall effects in graphene from proximity-induced uniform and staggered spin-orbit and exchange coupling. Phys. Rev. Lett., 124(13). 136403. https://doi.org/10.1103/physrevlett.124.136403 ) or by bringing it in double proximity between a layer of Cr$_2$Ge$_2$Te$_6$ and WS$_2$ ( Citation: Dolui & al., 2020 Dolui, K., Petrović, M., Zollner, K., Plecháč, P., Fabian, J. & Nikolić, B. (2020). Proximity spin–orbit torque on a two-dimensional magnet within van der Waals heterostructure: Current-driven antiferromagnet-to-ferromagnet reversible nonequilibrium phase transition in bilayer CrI3. Nano Lett., 20(4). 2288–2295. https://doi.org/10.1021/acs.nanolett.9b04556 ).

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