Electrical tuning of the magnetic properties of 2D magnets: the case of ${\rm Cr}_2{\rm Ge}_2{\rm Te}_6$
Abstract: Motivated by growing interest in atomically-thin van der Waals magnetic materials, we present an {\it ab initio} theoretical study of the dependence of their magnetic properties on the electron/hole density $\rho$ induced via the electrical field effect. By focusing on the case of monolayer ${\rm Cr}_2{\rm Ge}_2{\rm Te}_6$ (a prototypical 2D Ising ferromagnet) and employing a hybrid functional, we first study the dependence of the gap and effective mass on the carrier concentration $\rho$. We then investigate the robustness of magnetism by studying the dependencies of the exchange couplings and magneto-crystalline anisotropy energy (MAE) on $\rho$. In agreement with experimental results, we find that magnetism displays a bipolar electrically-tunable character, which is, however, much more robust for hole ($\rho>0$) rather than electron ($\rho<0$) doping. Indeed, the MAE vanishes for an electron density $\rho\approx - 7.5 \times 10{13}~{\rm e} \times {\rm cm}{-2}$, signalling the failure of a localized description based on a Heisenberg-type anisotropic spin Hamiltonian. This is in agreement with the rapid increase of the coupling between fourth-neighbor atoms with increasing electron density.
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- The RKKY interaction between two localized moments can be written as ℋRKKY=−A2𝑺1𝑺2χ~(𝑹12)/2subscriptℋRKKYsuperscript𝐴2subscript𝑺1subscript𝑺2~𝜒subscript𝑹122{\cal H}_{\rm RKKY}=-A^{2}{\bm{S}_{1}}{\bm{S}_{2}}{\widetilde{\chi}}({\bm{R}_{% 12}})/2caligraphic_H start_POSTSUBSCRIPT roman_RKKY end_POSTSUBSCRIPT = - italic_A start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT bold_italic_S start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT bold_italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT over~ start_ARG italic_χ end_ARG ( bold_italic_R start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT ) / 2. Here, A𝐴Aitalic_A is an interaction constant, 𝑹12subscript𝑹12{\bm{R}_{12}}bold_italic_R start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT is the distance between the localized spins, and χ~(𝑹12)~𝜒subscript𝑹12{\widetilde{\chi}}({\bm{R}_{12}})over~ start_ARG italic_χ end_ARG ( bold_italic_R start_POSTSUBSCRIPT 12 end_POSTSUBSCRIPT ) is the Fourier transform of the static density-density response function χ(𝒒)𝜒𝒒\chi({\bm{q}})italic_χ ( bold_italic_q ) (also known as “Lindhard function” [51]). In the present case, we have employed the density-density of a uniform non-interacting 2D electron gas [51], using mh*subscriptsuperscript𝑚hm^{*}_{\rm h}italic_m start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_h end_POSTSUBSCRIPT as band mass.
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