Magnetoelectric effect of multiferroic metals

Abstract

Much is known about the magnetoelectric effect of multiferroic insulators, yet little is understood about multiferroic metals. In this work, we employ first-principles calculations to identify the sliding van der Waals bilayer $1T$-NbTe$_2$ as a multiferroic metal, where in-plane metallicity coexists with out-of-plane polarization and magnetism. It exhibits linear magnetoelectric response, originating from direct spin-charge interactions as a result of external field-modulated Fermi energy, which differs from the spin-charge-lattice or spin-orbit coupling mechanisms in multiferroic insulators. We derive a universal formula for magnetoelectric coupling parameters of multiferroic metals, which highlights the crucial role of interlayer dielectric permittivity in enhancing performance. Our work provides insights for exploring magnetoelectric coupling mechanisms and designing functional materials with strong magnetoelectric coupling.

Figures (4)

• Figure 1: (a) Geometric structures for the ME state (Left), antiparallel AA' stacking (Middle), and $-$ME state of 1$T$-NbTe$_2$ bilayer. In the left and right panels, the black arrows connecting the orange and purple rhombuses denote the interlayer sliding vector l = $m$a + $n$b, with ($m$, $n$) = $(\frac{1}{3},\frac{2}{3})$ and $(\frac{2}{3},\frac{1}{3})$, respectively. $h$ and $d_z$ represent the bilayer thickness and the interlayer spacing, respectively. In the middle panel, $M_z$ denotes mirror symmetry between the top and down layers. Green and yellow balls denote Nb and Te atoms, respectively. (b) Total energy, (c) ferroelectric polarization, and (d) total magnetic moment as a function of ($m$, $n$) for the sliding 1$T$-NbTe$_2$ bilayer. In (c), the black dashed line indicates the ferroelectric switching path and the grey dot marks the transition state position.
• Figure 2: Spin-resolved band structure for the ME state, with spin-majority in red solid lines and spin-minority in blue dashed lines, respectively. The Fermi level is set at energy zero.
• Figure 3: Variation of (a) total magnetic moment ($M$) with applied electric field ($E$), and (b) polarization ($P$) with applied magnetic field ($\mu_0 H$), excluding and including spin-orbit coupling (w/o and w/ SOC). Data points are calculated from first-principles calculations, with curves representing linear fits. Strain-dependent (c) total magnetic moment and polarization, and (d) magnetoelectric coupling parameters $\alpha^{E}$ and $\alpha^{H}$.
• Figure 4: Schematic of the Fermi-energy-modulation mechanism for the magnetoelectric response of the 1$T$-NbTe$_2$ bilayer. In the upper row, red and blue represent the electron states of spin-up and spin-down, respectively. In the lower row, red-upward and blue-downward arrows represent electrons carrying spin-up and spin-down, while green arrows indicate interlayer charge transfer. Black-dashed boxes represent the electrons being transferred out and the black-solid boxes represent the electrons being received. Black-curved arrows indicate the spin flipping within the layer under a vertical magnetic field. Middle panel. For the high-symmetry AA' configuration, no interlayer charge transfer exists due to the $M_z$ symmetry. Although the monolayer NbTe$_2$ exhibits a spin-splitting $\Delta$, it is of equal magnitude but opposite sign between the top and down layers, forming an antiferromagnet with a zero total moment. $E_F$ denotes the position of the Fermi level. Left panel. An applied $z$-orientated electric field induces an interlayer potential difference, causing the electronic states of the top layer to shift upwards relative to the down layer and resulting in a difference in individual Fermi levels ($E^T_F$ vs. $E^D_F$ in the top-left). To maintain an identical Fermi level across the system, a portion of electrons from the top layer transfer to the down layer. This process is accompanied by spin flipping, preventing the interlayer magnetic moments from fully compensating and modifying the total moment. Right panel. An applied $z$-orientated magnetic field produces two effects. On the one hand, it excites some spins to flip towards the direction of the external magnetic field. On the other hand, it introduces a Zeeman splitting, $\Delta_H$, which broadens the spin-splitting in the top layer and narrows it in the down layer due to antiparallel interlayer coupling. This asymmetry creates a difference in Fermi levels between layers, thereby driving interlayer charge transfer and altering the polarization.