Oxidation tuning of ferroic transitions in Gd₂C monolayer

As shown in Fig. 1, in Gd₂C monolayer, the carbon layer is sandwiched between two gadolinium layers. The sublattices of both C and Gd possess the trigonal geometry, and the space group (S.G.) of Gd₂C monolayer is $P\bar{3}m1$. Its dynamical stability has been confirmed by phonon calculation, and no imaginary mode appears in the phonon spectrum over the entire Brillouin zone, as shown in Fig. S2 of SM.

To determine Gd₂C monolayer’s magnetic ground state, four most possible magnetic orders are compared, as shown in Figs. 1(c-f). According to our calculation, the energy of FM is significantly lower than those of other three (as summarized in Table 1), implying a FM ground state, consistent with its bulk property. [23] Our calculation shows a large total magnetization ($7.96$ $\mu_{\rm B}$/Gd), very close to the ideal 8 $\mu_{\rm B}$/Gd²⁺ for the 4$f^{7}$+5$d^{1}$ configuration. Besides, our DFT result is also consistent with a previous study. [25] And the local magnetic moment of Gd ion is $7.38$ $\mu_{\rm B}$, smaller than $8$ $\mu_{\rm B}$, which is also reasonable. The local magnetic moment in VASP calculation is integrated within the default Wigner-Seitz sphere of atom/ion. For Gd²⁺ with a spatially-extended $5d$ electron, numerically, it is natural to obtain a relative smaller local magnetic moment here. The electronic structure of Gd₂C monolayer suggests a robust metallic state (Fig. S3 in SM), despite the magnetic orders.

For MXenes, their surfaces are usually passivated by halogen, oxygen, or some groups. Here we first consider the fluorine atoms as the surface coverage, which can oxidize Gd₂C. For Gd₂CF₂ monolayer, there are five most possible sites for F adatoms, [46, 47, 48] as shown in Fig. S4 in SM. According to our calculations, the model 2 of adsorption is the most stable structure, which has the lowest energy among all considered configurations (see Fig. S5(a) in SM). There is no apparent imaginary vibration mode in its phonon spectrum, as shown in Fig. S5(b) of SM, indicating its dynamic stability. And the local magnetic moment is reduced to $\approx 7$ $\mu_{\rm B}$, implying Gd³⁺ as expected. The ground state is also changed to the Z-AFM one. Figure 2(a) shows its atomic- projected density of states (DOS), which suggests a moderate band gap ($1.1$ eV). Near the Fermi level, it is evident that there is covalent hybridization between the Gd’s $5d$ and C’s $2p$ orbitals, although nominally the $5d$ orbitals should be empty in the ionic crystal limit for Gd³⁺. The carbon ion, accommodating enough electrons with the close-shell $2p^{6}$ configuration, keep a symmetrical centeral position stably.

Besides fluorine, oxygen can be also used to passive the surface of Gd₂C monolayer, which can draw one more electron. Five structural models with four possible magnetic orders are also tested, as done in above fluorine case. Differently, the polar model 3 of adsorption has the lowest energy among all these structures, as shown in Fig. S6(a) of SM. There is no apparent imaginary vibration mode, as shown in Fig. S6(b) of SM, indicating its dynamic stability. The Z-AFM phase remains the ground state in Gd₂CO₂, as in Gd₂CF₂ monolayer. The local magnetic moment of Gd in Gd₂CO₂ remains $\approx 7$ $\mu_{\rm B}$, implying Gd³⁺. And its DOS is similar to the F-passived case, with a moderate band gap ($1.1$ eV), as shown in Fig. 2(b). Then the nominal valence of C must be $-2$ in Gd₂CO₂.

The most interesting result is that ferroelectricity emerges in Gd₂CO₂ monolayer, and thus it becomes a 2D multiferroic system. Its polarization reaches $18.23$ pC/m ($4.53$ $\mu$C/cm² in the 3D form), larger than those of WTe₂ monolayer ($0.11$ $\mu$C/cm²) and CuCrP₂S₆ ($0.79$ pC/m). [49, 50] The polarization is mainly along the out-of-plane direction, although its $Cm$ symmetry from ZAFM order allows an in-plane component (negligible $\sim$0.28 pC/m). Therefore, only the out-of-plane polarization is studied in the following.

The ferroelectric switching energy barrier is also estimated in Fig. 2(c), which reaches $0.7$ eV/f.u., smaller than that of GaFeO₃ ($1.05$ eV/f.u.) and comparable to vacancy-induced dipole switching in CrI₃ monolayer ($0.65$ eV/f.u.), [51, 52] which suggests the stability and switchability of ferroelectric phase.

The electronic clouds of Gd₂CF₂ and Gd₂CO₂ monolayers are compared in Figs. 3(a-b), which can help the understanding of ferroic transitions. For Gd₂CF₂, electrons are evenly distributed. For Gd₂CO₂, there is electronic overlap between the top oxygen and carbon, implying the C-O bonding. Such coordinate bonding is due to the hybridization between C’s $2p$ and O’s $2p$ orbitals, leading to the unidirectional displacement of middle C^2- ion towards upper or lower oxygen. Such an origin of ferroelectricity is exotic, which enlarges the scope of ferroelectric materials.

In principle, the full coverage of fluorine can be also mimicked by $50\%$ coverage of oxygen, at least in the terms of valence. Thus the ferroic phase of monolayer Gd₂CO_δ can transform from a ferromagnetic metal ($\delta=0$) to an antiferromagnetic insulator ($\delta=1$) accompanying by a change from Gd²⁺ to Gd³⁺, then to a multiferroic insulator ($\delta=2$) with C^4- transforming to C^2-. Therefore, the fine tuning of O’s concentration could manipulate the tri-ferroic-state phase transitions, as illustrated in Fig. 3(c). The tuning of oxidation may be operated via ionic liquid gating. [13] When the voltage gating across the liquid is negative, the negative internal electric field can drive O^2- ions into the material, and vise versa, extract O^2- ions by positive electric field.

In practice, the optical methods are convenient (sensitive and nondestructive) to detect the ferroicity in 2D materials. For example, the MOKE has been used as a powerful tool for the characterization of the low-dimensional magnetic materials, [4, 5, 6] and the SHG has been widely employed to characterize those materials lacking inversion symmetry. [53, 54] Hence, they may be the proper techniques to distinguish these tri-state phases.

The magnetic point group of the Gd₂C monolayer is $-3m^{\prime}$, and the corresponding optical conductivity can be expressed as Eq. S1 in the SM. [55] Generally, the signal of MOKE is associated with the off-diagonal components of optical conductive tensor $\sigma$. Therefore, the MOKE signal can be obtained in the Gd₂C monolayer, charactered by the complex Kerr angle $\phi_{\rm K}=\theta_{\rm K}+i\eta_{\rm K}$, where $\theta_{\rm K}$ is the Kerr rotation angle and $\eta_{\rm K}$ is the Kerr ellipticity, as shown in Fig. 4(a), more details can be found in SM. However, no MOKE signal can be detected in the antiferromagnetic Gd₂CO₂ and Gd₂CF₂ monolayers. Although their time reversal symmetry ($\hat{T}$) is naturally broken by magnetic order, the $\hat{T}\hat{t}$ symmetry ($\hat{t}$ is the half unit cell translation) remains unbroken.[56]

The Gd₂C and Gd₂CF₂ monolayers belong to the space group $P\bar{3}m1$ and $C2/m$ respectively, both of which are spatial inversion symmetric. Hence, there is no SHG response for these two monolayers. In contrast, the space inversion symmetry is broken in Gd₂CO₂ monolayer with the space group $Cm$, which allows the intrinsic SHG signal. Here, we use the orthorhombic cell to calculate the SHG of Gd₂CO₂ monolayer, as shown in Fig. 4(b). And the nonlinear optical susceptibility tensors ($d_{ij}$) of Gd₂CO₂ monolayer can be expressed as a $3\times 6$ matrix: [57]

The calculated $d_{ij}$’s of Gd₂CO₂ monolayer are listed in Fig. 4(c). Although the calculated values of $d_{ij}$’s depend on the vacuum thickness (see Fig. S7 in SM), their magnitudes are unaffected semiquantitatively. For comparison, the $d_{ij}$’s of KH₂PO₄ (KDP, a frequently-used reference of nonlinear optical materials) with space group $I\bar{4}2d$ are also calculated. Our calculated result is in consistent with the experimental value ($d_{36}$=0.38 pm/V), [58] implying the reliability of our SHG calculations. The dominant components of Gd₂CO₂ monolayer, i.e., $d_{15}$ and $d_{31}$, are much larger than those of KDP.

For Gd₂CO₂ monolayer under perpendicular incident light, the SHG intensity can be expressed as: [57, 59] $I\propto(2d_{26}\sin 2\phi)^{2}+(2d_{11}\cos^{2}\phi+2d_{12}\sin^{2}\phi)^{2}+(2d_{31}\cos^{2}\phi+2d_{32}\sin^{2}\phi)^{2}$, where $\phi$ is the angle between the crystalline $a$-axis and the electric field direction of incident light, as explained in SM. The calculated SHG angular plot is shown in Fig. 4(d), which shows the dumbbell shape and much stronger than that of KDP.

In short, the combination of MOKE and SHG techniques can be used to monitor the ferroic tri-state transitions in Gd₂CX₂.