Realization of anomalous multiferroicity in free-standing graphene with magnetic adatoms

It is generally believed that free-standing graphene does not demonstrate any ferroic properties. In the present work we revise this statement and show that a single graphene sheet with a pair of magnetic adatoms can be driven into ferroelectric (FE) and multiferroic (MF) phases by tuning the Dirac cones slope. The transition into the FE phase occurs gradually, but an anomalous MF phase appears abruptly by means of a quantum phase transition. Our findings suggest that such features should exist in graphene recently investigated by scanning tunneling microscopy [H. González-Herrero et al., Science 352, 437 (2016)].

Figure 1

MF phase of two energetically different magnetic adatoms ($\mathrm{\Delta }\mathcal{E}$) collinear to a carbon: distinct charge accumulations ${\delta }^{+}$ and ${\delta }^{-}$ together with a net magnetization (vertical arrows) can split over these adatoms. In this system, a QPT modifies abruptly the FE phase into the MF, due to the increasing of the Fano factor $q_0>q_c$ above the critical point, via the tuning of the Dirac cones slope (the Fermi velocity $v_F$).

Figure 2

(a) Magnetization of the system as a function of the Fano factor $q_0$ for fixed detuning $\mathrm{\Delta }\varepsilon$. At critical value of the parameter $q_c$ the system is driven to the MF phase by a QPT. (b) The sketch illustrating the connection between the slope of the Dirac cones and magnetization of the system. (c) Dipole moment of the system as a function of the Fano factor $q_0$ for fixed detuning $\mathrm{\Delta }\varepsilon$. After slow initial increase, the dipole moment experiences fast growing, followed by slow decrease and discontinuity at $q_0=q_c$. Rising of an anomalous MF phase for $q_0>q_c.$ (d) Magnetization of the system as a function of the detuning $\mathrm{\Delta }\varepsilon$ for fixed Fano factor $q_0$. (e) Dipole moment of the system as a function of the detuning $\mathrm{\Delta }\varepsilon$ for fixed $q_0$. The MF phase exists within the critical region $\mathrm{\Delta }{\varepsilon }_{\text{c}1}<\mathrm{\Delta }\varepsilon <\mathrm{\Delta }{\varepsilon }_{\text{c2}},$ except for the FM phase where $\mathrm{\Delta }\varepsilon =0.$ Otherwise, just the FE phase is present.

Figure 3

(a)–(b) Magnetizations of the adatoms as a function of both $q_0$ and $\mathrm{\Delta }\varepsilon$. (c) The dipole moment of an adatom pair as a function of both $q_0$ and $\mathrm{\Delta }\varepsilon$. (d) Top plot: phase diagram for FM order parameter; middle plot: phase diagram for FE order parameter; and bottom plot: total phase diagram for N, FE, FM, and MF phases.

Figure 4

(a)–(b) Spin-degenerate and charge split DOSs of the adatoms, due to the finite dipole moment, representing the FE phase, which can be probed by an STM tip. (c)–(d) In the MF phase, the DOSs are simultaneously charge and spin split.