Orbital longitudinal magnetoelectric coupling in rhombohedral multilayer graphene

Significance

In multiferroic materials, magnetoelectric coupling has attracted significant interest for enabling the manipulation of magnetization by electric fields and vice versa. We present a microscopic theory of orbital longitudinal magnetoelectric coupling (LMC) in van der Waals heterostructures, with particular focus on rhombohedral multilayer graphene systems. Our analysis reveals that LMC is electrically tunable through gate control and originates fundamentally from a layer-space quantum geometry. This theoretical framework provides both quantitative understanding for LMC, as well as the emergent magnetoelectric properties of rhombohedral multilayer graphene.

Abstract

Magnetoelectric coupling enables the manipulation of magnetization by electric fields and polarization by magnetic fields. While typically found in heavy element materials with large spin–orbit coupling, recent experiments on rhombohedral-stacked pentalayer graphene have demonstrated a longitudinal magnetoelectric coupling (LMC) without spin–orbit coupling. Here, we develop a microscopic theory of LMC in layered quantum materials and identify how it is controlled by a “layer-space” quantum geometry. Focusing on rhombohedral multilayer graphene systems, we find that the interplay between LMC and valley-polarized order produces a butterfly shaped magnetic hysteresis controlled by out-of-plane electric field: a signature of LMC and a multiferroic valley order. Furthermore, we identify a nonlinear LMC in rhombohedral multilayer graphene under time-reversal symmetry, while the absence of centrosymmetry enables the generation of a second-order nonlinear electric dipole moment in response to an out-of-plane magnetic field. Our theoretical framework provides a quantitative understanding of LMC, as well as the emergent magnetoelectric properties of rhombohedral multilayer graphene.

Multilayer graphene has emerged as a versatile van der Waals heterostructure for controlling electronic behavior (1–3). A variety of exotic phenomena have been observed in multilayer graphene that include interaction-driven Chern insulators (4–6), ferromagnetism (7, 8), and superconductivity (9, 10). In rhombohedral-stacked pentalayer graphene (RPG) multiferroicity (11) has been recently reported, manifesting a magnetic ordering that can switched by gate electric field. This phenomenon is particularly striking since spin–orbit coupling in RPG vanishes. Instead, such multiferroicity arises hand-in-hand with an orbital magnetoelectric coupling that mediates the transduction between electric and magnetic fields within materials. This transduction enables the control of magnetization (or polarization) through electric (or magnetic) fields (12–21).

In the modern theory of magnetoelectric coupling, major attention has focused on axion electrodynamics (22, 23), where the axion coupling is half quantized in units of $e^2/2h$ (24–30). Besides the topological magnetoelectric effects, spin–orbit coupling can also induce linear magnetoelectric coupling in three-dimensional materials (31). Furthermore, in the current-carrying state, the gyrotropic magnetoelectric effect is determined by the spin and orbital magnetic moments on the Fermi surface (32). In contrast to the above seminal works, the emergent magnetoelectric coupling in 2D layered quantum materials found recently in the experiment of ref. 11 can persist even in the absence of spin–orbit coupling (33).

Here, we argue that the longitudinal magnetoelectric coupling (LMC) in 2D layered materials are mediated by a layer-dependent quantum geometry manifesting as an out-of-plane control of magnetization by a gate electric field, and vice versa. In particular, we unveil a microscopic orbital magnetoelectric moment that emerges from examining how the magnetic moment changes with an applied out-of-plane electric field as well as how layer dipole moment changes with an applied magnetic field (Fig. 1). Critically, as we discuss below, valley polarization, a unique correlated order in multilayer graphene, plays a central role in the linear LMC observed in RPG by lifting the valley degeneracy of its electronic bands. Notably, we find that LMC can intertwine with the valley-polarized order yielding an “electric field-valley polarization” coupling. As we explain below this naturally produces a butterfly-shaped hysteresis recently observed in experiments (11). When valley polarization is turned off, we find layer mediated LMC can persist as nonlinear LMC. For example, such nonlinear LMC allows magnetic field to control layer polarization for noncentrosymmetric but time-reversal invariant systems. As an intrinsic effect, nonlinear LMC generally exists not only in multilayer graphene, but also in other van der Waals layered materials and includes contributions from both the Fermi-surface and the Fermi-sea terms. These highlight how the rich orbital magnetoelectric effects in layered materials is mediated by the layer degree of freedom.

Fig. 1.

(A) Schematic illustration of the rhombohedral pentalayer graphene embedded in a dual-gate device. With the aid of Top and Bottom gates, both the carrier density and out-of-plane electric field can be tuned. The distance between the Top and Bottom layers is d. (B) The layer-polarized orbital magnetic moment acquires a correction in response to E field, while the layer electric dipole moment receives a correction in response to the B field. Both of them contribute to an emergent orbital magnetoelectric moment described in Eq. 10.

Results and Discussion

Orbital Magnetoelectric Moment & Linear LMC in Multilayers.

Microscopically, the LMC can be generally described via:

$$\begin{array}{c}\hfill \delta M={\chi }_{\mathit{me}}E+{\gamma }_e{E}^2+...\end{array}$$ [1]

$$\begin{array}{c}\hfill \delta P={\chi }_{\mathit{em}}B+{\gamma }_b{B}^2+...,\end{array}$$ [2]

where $\delta M$ is the induced magnetization, $\delta P$ is the polarization change, and E and B are the applied electric and magnetic fields respectively; in what follows we will focus on the case where all fields, polarization, and magnetization are out-of-plane. Here, the linear magnetoelectric coefficients are denoted ${\chi }_{\mathit{me}}$ and ${\chi }_{\mathit{em}}$ respectively; a maxwell relations ensures ${\chi }_{\mathit{em}}={\chi }_{\mathit{me}}$. Importantly, in order for nonzero ${\chi }_{\mathit{me}}$ and ${\chi }_{\mathit{em}}$ to manifest, time reversal, inversion, and mirror symmetries need to be broken. In contrast, ${\gamma }_e$ survives centrosymmetry and ${\gamma }_b$ survives time reversal.

To evaluate the longitudinal electric and magnetic responses in layered quantum materials, we adopt a semiclassical approach that incorporates geometric phases and orbital magnetization. We begin by analyzing an effective field-dependent free energy $F(E,B)$ as (34, 35)

$$\begin{array}{c}\hfill F(E,B)=-\frac{1}{\beta }{\displaystyle \sum _{n\mathit{k}}}\{1+\frac{e}{ħ}B{\tilde{\mathrm{\Omega }}}_{n\mathit{k}}\}\varphi [{\epsilon }_{n\mathit{k}}(E,B)],\end{array}$$ [3]

where $n,\mathit{k}$ are the band index and wavevector, $\varphi (\epsilon )=\ln [1+e^{-\beta (\epsilon -\mu )}]$. We model the field-dependent Bloch band electronic energy via ${\epsilon }_{n\mathit{k}}(E,B)={\epsilon }_{n\mathit{k}}^{(0)}+\delta {\epsilon }_{n\mathit{k}}(E,B)$. Here, ${\epsilon }_{n\mathit{k}}^{(0)}$ is the bare Bloch band electronic energy without applied E or B fields, ${\tilde{\mathrm{\Omega }}}_{n\mathit{k}}$ is the Bloch band Berry curvature, $\beta =1/k_{B}T$ is an inverse temperature, and μ is the chemical potential. $\delta {\epsilon }_{n\mathit{k}}(E,B)$ are field-induced corrections discussed in detail below. In the following, spin acts as a spectator degree of freedom since graphene systems possess negligible spin–orbit coupling.

As we now argue, the layer degree of freedom mediates an emergent orbital magnetoelectric coupling in multilayer graphene. This can be revealed by examining how the Bloch electronic states change under an applied out-of-plane electric field E and magnetic field B via

$$\begin{array}{c}\hfill H(E,B)=H_0(\mathit{k})+U\widehat{\mathit{p}}-\widehat{\mathit{m}}B,\end{array}$$ [4]

where U is the layer potential drop induced by electric field and d is the distance between Top and Bottom layers (Fig. 1A). Here, $H_0(\mathit{k})$ is the bare material Hamiltonian such that $H_0(\mathit{k})|n\mathit{k}⟩={\epsilon }_{n\mathit{k}}^{(0)}|n\mathit{k}⟩$ with $|n\mathit{k}⟩$ the Bloch eigenstate, $\widehat{\mathit{p}}$ is the layer-polarization operator, which is generally given by

$$\begin{array}{c}\hfill \widehat{\mathit{p}}={\displaystyle \sum _{\ell ,\alpha }}(\ell /d)|{\varphi }_{\ell ,\alpha }⟩⟨{\varphi }_{\ell ,\alpha }|,\end{array}$$ [5]

where $|{\varphi }_{\ell ,\alpha }⟩$ denotes the α orbital on the layer at height ℓ measured from the center of the stack. For example, for a bilayer system $\ell =(d/2)\{-1,1\}$ and $\widehat{\mathit{p}}={\sigma }_z/2$ (36, 37). $\widehat{\mathit{m}}=e(\widehat{\mathit{v}}\times \widehat{\mathit{r}}-\widehat{\mathit{r}}\times \widehat{\mathit{v}})/4$ is the orbital magnetic momentum operator (38) that describes an orbital-Zeeman coupling with $ħ\widehat{\mathit{v}}={\mathrm{\nabla }}_{\mathit{k}}{H}_0(\mathit{k})$ the velocity operator and $\widehat{\mathit{r}}$ the position operator. By directly expanding the energy of the Bloch states $|n\mathit{k}⟩$ to order $\mathcal{O}(UB)$ we find the field-corrected band electronic energy reads as

$$\begin{array}{c}\hfill \delta {\epsilon }_{n\mathit{k}}(E,B)=p_{n\mathit{k}}U-m_{n\mathit{k}}B+{\alpha }_{n\mathit{k}}UB,\end{array}$$ [6]

where $p_{n\mathit{k}}=⟨n|\widehat{\mathit{p}}|n⟩$ is the layer dipole moment for band n and $m_{n\mathit{k}}=(e/2)\sum _{m\ne n}{\mathit{v}}_{\mathit{nm}}\times {\mathit{r}}_{\mathit{mn}}$ is the intraband orbital moment with ${\mathit{v}}_{\mathit{nm}}$ and ${\mathit{r}}_{\mathit{nm}}$ the interband velocity matrix elements and interband Berry connection respectively.

While the first two terms in Eq. 6 represent the Stark shift and orbital Zeeman effect arising directly from the first-order expectation value, the last term in Eq. 6${\alpha }_{n\mathit{k}}$ describes an orbital magnetoelectric moment, emerging from a layer-dependent quantum geometry. To see this, notice that $U\widehat{\mathit{p}}$ perturbs the Bloch eigenstates $|n\mathit{k}⟩$. To first order in electric fields, the velocity matrix elements change as ${\tilde{\mathit{v}}}_{\mathit{nm}}\to {\mathit{v}}_{\mathit{nm}}+U{\delta }_u{\mathit{v}}_{\mathit{nm}}$, where

$$\begin{array}{c}\hfill {\delta }_u{\mathit{v}}_{\mathit{nm}}={\displaystyle \sum _{q\ne n}}{p}_{\mathit{nq}}{\mathit{v}}_{\mathit{qm}}/{\epsilon }_{\mathit{nq}}^{(0)}+{\displaystyle \sum _{l\ne m}}{p}_{\mathit{lm}}{\mathit{v}}_{\mathit{nl}}/{\epsilon }_{\mathit{ml}}^{(0)},\end{array}$$ [7]

where ${\epsilon }_{\mathit{mn}}^{(0)}={\epsilon }_{m\mathit{k}}^{(0)}-{\epsilon }_{n\mathit{k}}^{(0)}$. In the same fashion, the changes to the interband Berry connections can be computed from the continuity equation ${\mathit{r}}_{\mathit{mn}}^a=⟨m|i{\partial }_a|n⟩=-iħ{\mathit{v}}_{\mathit{mn}}^a/{\epsilon }_{\mathit{mn}}$ (for $m\ne n$); see SI Appendix, section I for details. Critically, ${\delta }_u$ produces layer potential induced changes to the orbital magnetic moment and its Berry curvature: ${\tilde{m}}_{n\mathit{k}}\to m_{n\mathit{k}}+U{\delta }_u{m}_{n\mathit{k}}$ and ${\tilde{\mathrm{\Omega }}}_{n\mathit{k}}\to {\mathrm{\Omega }}_{n\mathit{k}}+U{\delta }_u{\mathrm{\Omega }}_{n\mathit{k}}$. The magnetic moment polarizability ${\delta }_u{m}_{n\mathit{k}}$ and Berry curvature polarizability ${\delta }_u{\mathrm{\Omega }}_{n\mathit{k}}$ can be obtained as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\delta }_u{m}_{n\mathit{k}}=-{\displaystyle \sum _{m\ne n}}\frac{eħ}{{[{\epsilon }_{\mathit{nm}}^{(0)}]}^2}& \mathrm{Im}[{\epsilon }_{\mathit{nm}}^{(0)}{\delta }_u{\mathit{v}}_{\mathit{nm}}\times {\mathit{v}}_{\mathit{mn}}+\hfill \\ \hfill & v_{\mathit{nm}}^x{v}_{\mathit{mn}}^y(p_m-p_n)],\hfill \end{array}\end{array}$$ [8]

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\delta }_u{\mathrm{\Omega }}_{n\mathit{k}}=-{\displaystyle \sum _{m\ne n}}\frac{2{ħ}^2}{{[{\epsilon }_{\mathit{nm}}^{(0)}]}^3}& \mathrm{Im}[{\epsilon }_{\mathit{nm}}^{(0)}{\delta }_u{\mathit{v}}_{\mathit{nm}}\times {\mathit{v}}_{\mathit{mn}}+\hfill \\ \hfill & 2v_{\mathit{nm}}^x{v}_{\mathit{mn}}^y(p_m-p_n)],\hfill \end{array}\end{array}$$ [9]

where we have recalled ${\mathrm{\Omega }}_{n\mathit{k}}=i\sum _{m\ne n}{\mathit{r}}_{\mathit{nm}}\times {\mathit{r}}_{\mathit{mn}}$. Notice that Eq. 8 and 9 describes how time-reversal odd quantum geometric quantities change with layer electric potential difference. Such layer potential–dependent quantum geometry is possible due to tunneling between layers in $H_0(\mathit{k})$: It vanishes when the layers are decoupled.

Using Eq. 7 to 9 and collecting terms $\mathcal{O}(UB)$ to the energy $⟨H(E,B)⟩$, we obtain an explicit form of the orbital magnetoelectric moment as

$$\begin{array}{c}\hfill {\alpha }_{n\mathit{k}}=-{\delta }_u{m}_{n\mathit{k}}+2{\displaystyle \sum _{l\ne n}}\frac{\mathrm{Re}(p_{\mathit{nl}}{m}_{\mathit{ln}})}{{\epsilon }_{\mathit{ln}}^{(0)}},\end{array}$$ [10]

where first term in Eq. 10 can be traced to changes to the Zeeman energy with respect to E field. The second term in Eq. 10 arises from a B field induced change to the layer dipole energy. Eq. 10 is one of the key results of our work and reveals how electrons can intrinsically possess a magnetoelectric coupling.

Interestingly, both the field induced changes to layer dipole energy and Zeeman energy contribute to variations in the layer dipole and magnetic moment. In particular, the layer dipole and orbital magnetic moment satisfy a “microscopic” Maxwell relation, namely

$$\begin{array}{c}\hfill {\displaystyle \underset{U\to 0}{lim}}\frac{\partial ⟨{\psi }_{n\mathit{k}}^{\lambda }|\widehat{\mathit{p}}|{\psi }_{n\mathit{k}}^{\lambda }⟩}{\partial B}=-{\displaystyle \underset{B\to 0}{lim}}\frac{\partial ⟨{\psi }_{n\mathit{k}}^{\lambda }|\widehat{\mathit{m}}|{\psi }_{n\mathit{k}}^{\lambda }⟩}{\partial U}={\alpha }_{n\mathit{k}},\end{array}$$ [11]

where $|{\psi }_{n\mathit{k}}^{\lambda }⟩$ is the eigenstate of $H(E,B)$ with the parameter $\lambda =U,B$. We have used the Feynman–Hellmann theorem together with Eq. 6. Note that the interband orbital moment $m_{n{n}^{\prime }}$ vanishes for a two-band model zeroing the second term (dipole energy change) in Eq. 10. Even in this limit, the microscopic Maxwell relation in Eq. 11 is still satisfied: Counterintuitively, the magnetic field induced layer dipole depends on E field induced changes to a Zeeman energy. This further demonstrates how the layer dipole is intimately intertwined with the magnetic moment.

Moving to the total induced magnetization and polarization in a sample, we write $\delta M=-\underset{B\to 0}{lim}{\partial }_{B}F(E,B)$ and $\delta P=-\underset{E\to 0}{lim}{\partial }_{E}F(E,B)$ to directly obtain the linear LMC coefficients from Eq. 3 as

$$\begin{array}{cc}\hfill {\chi }_{\mathit{me}}& ={\chi }_{\mathit{em}}=ed{\displaystyle \sum _{n\mathit{k}}}\{-{\alpha }_{n\mathit{k}}{f}_{n\mathit{k}}+m_{n\mathit{k}}{p}_{n\mathit{k}}{f}_{n\mathit{k}}^{\prime }\hfill \\ \hfill & +(e/ħ)({\delta }_u{\mathrm{\Omega }}_{n\mathit{k}}\varphi [{\epsilon }_{n\mathit{k}}^{(0)}]/\beta -{\mathrm{\Omega }}_{n\mathit{k}}{p}_{n\mathit{k}}{f}_{n\mathit{k}})\},\hfill \end{array}$$ [12]

where $f_{n\mathit{k}}=-{\partial }_{ϵ}\varphi (ϵ)/\beta =\{1+\exp [\beta ({\epsilon }_{n\mathit{k}}^{(0)}-\mu ){]\}}^{-1}$ is the Fermi–Dirac distribution function. Here, we have used $U=edE$. It was also pointed out that in multilayer graphene, the density imbalance between $A_1$ and $B_N$ causes a screening field, which reduces the layer energy drop (39). Due to the screening effect, the layer potential U and applied electric field E can be related by $U=edh(E)$, and at weak electric fields, $h(E)=\mathrm{\Lambda }E$ with a renormalized factor Λ ($\mathrm{\Lambda }<1$). Incorporating this screening effect, the magnetoelectric susceptibility ${\chi }_{\mathit{me}}$ can be renormalized as $\mathrm{\Lambda }{\chi }_{\mathit{me}}$. Since the m and Ω terms are time-reversal-odd, ${\delta }_u$ and p terms are inversion-odd, both time reversal and inversion symmetries need to be broken to produce a nonvanishing ${\chi }_{\mathit{me}}$. The quantities ${\delta }_u{\mathrm{\Omega }}_{n\mathit{k}}$ and ${\alpha }_{n\mathit{k}}$ are intrinsic, endowed by the quantum-geometric nature of Bloch electrons in the layer space. Since Eq. 12 provides a general description of LMC, we know that topological antiferromagnetic thin films manifest a half quantized topological magnetoelectric effect with ${\chi }_{\mathit{me}}=e^2/2h$, as detailed in SI Appendix, section IV. The primary focus of this work is the linear LMC observed in rhombohedral multilayer graphene systems, particularly in RPG.

To be concrete, in the following we focus on the linear LMC coefficients in multilayer graphene systems, for example, as observed in RPG (11). The K and $K^{\prime }$ valleys possess opposite orbital magnetic moments and Berry curvature, as well as opposite ${\delta }_u{\mathrm{\Omega }}_{n\mathit{k}}$ and ${\alpha }_{n\mathit{k}}$. To facilitate our analysis, we will focus on a metallic state with a valley-polarized order parameter, ${\mathrm{\Delta }}_v$ (${\mathrm{\Delta }}_v\ll \mu ,W$) where W is the bandwidth. Such valley-polarized (metallic) magnetic order has been widely used to describe magnetic ordering in multilayer graphene via ${\epsilon }_n^{\tau }\to {\epsilon }_n^{\tau }+\tau {\mathrm{\Delta }}_v$ (40, 41), where $\tau =\pm 1$ denotes the valley index. The general formula presented in Eq. 12 can also be applied to study other correlated states through Hartree–Fock calculations (33, 42, 43).

Expanding the energies in Eq. 12, we directly find

$$\begin{array}{c}\hfill \frac{{\chi }_{\mathit{me}}}{2ed{\mathrm{\Delta }}_v}=-{\displaystyle \sum _{n\mathit{k}\in \mathrm{K}}}[({\alpha }_{n\mathit{k}}+\frac{e}{ħ}{\mathrm{\Omega }}_{n\mathit{k}}{p}_n)f_{n\mathit{k}}^{\prime }+\frac{e}{ħ}{\delta }_u{\mathrm{\Omega }}_{n\mathit{k}}{f}_{n\mathit{k}}].\end{array}$$ [13]

In obtaining Eq. 13, we have have summed over both valleys so that the momentum integral in Eq. 13 is only carried out within the K valley; we have also used Eq. 10 and suppressed terms going as $f^{″}$ since it is odd in energy. Importantly, Eq. 13 demonstrates that the LMC in valley-polarized layered metals depends on ${\mathrm{\Delta }}_v$ and have contributions that are delineated into Fermi-surface ($f_{n\mathit{k}}^{\prime }$) and Fermi-sea ($f_{n\mathit{k}}$) contributions. Note that in two-band models (widely used in describing multilayer graphene) the second term in Eq. 10 vanishes. Even for multiband model, the interband orbital moment in multilayer graphene is small (44), making ${\alpha }_{n\mathit{k}}\approx -{\delta }_u{m}_{n\mathit{k}}$ a good approximation of the magnetoelectric moment.

LMC in Rhombohedral Pentalayer Graphene.

To demonstrate LMC we now turn our attention to RPG as depicted by the crystal structure in Fig. 1A. Close to charge neutrality, the electron wave functions in RPG are nearly localized at A sites on the top layer and B sites for the bottom layer. Therefore, the low-energy electronic properties of RPG can be described by the continuum Hamiltonian acting in the pseudospin basis of ($A_1$, $B_5$), which reads (45)

$$\begin{array}{c}\hfill H_{\tau }(\mathit{k})=\frac{v_0^5}{{\gamma }_1^4}\left(\begin{array}{cc}0& {({\pi }^{†})}^5\\ {\pi }^5& 0\end{array}\right)+\frac{{\mathrm{\Delta }}_d}{2}{\sigma }_z+H_R,\end{array}$$ [14]

where $\pi =\tau p_x+i{p}_y$. The inversion breaking term ${\mathrm{\Delta }}_d$ is an adjustable interlayer potential difference between the top and bottom layers and $d=1.4$nm is the interlayer distance. $H_R$ is the remote hopping-term corrections. The lattice constant of graphene is $a_0=2.46$. Using Eq. 10, we plot the magnetoelectric moment $-{\alpha }_{n\mathit{k}}={\delta }_u{m}_{n\mathit{k}}$ (for the two band model in Eq. 14) and ${\delta }_u{\mathrm{\Omega }}_{n\mathit{k}}$ in the momentum space for the valence band of RPG in Fig. 2 A and B demonstrating how the magnetoelectric moment is concentrated at the band edges.

Fig. 2.

(A and B) The momentum-space ${\delta }_{u}m$ and ${\delta }_u\mathrm{\Omega }$ for the hole band of RPG at ${\mathrm{\Delta }}_d=4$ meV. (C) The LMC coefficient ${\chi }_{\mathit{me}}$ as a function of carrier density $n_e$ at the temperature $T=0.8$ K and ${\mathrm{\Delta }}_d=2$ meV. The Fermi-surface, Fermi-sea, and total contributions are plotted by blue, orange, and green colors, respectively. The spin degeneracy is considered in the calculation. (D) ${\chi }_{\mathit{me}}$ as a function of $n_e$ at different ${\mathrm{\Delta }}_d$.

We evaluate the linear LMC coefficient ${\chi }_{\mathit{me}}$ as a function of carrier density in Fig. 2C. Here, we have used ${\mathrm{\Delta }}_v=0.2$ meV estimated from the transition temperature for the onset of valley order in RPG (11). Evidently, ${\chi }_{\mathit{me}}$ is maximized near the carrier density $n_e\approx -0.4\times {10}^{12}{\mathrm{cm}}^{-2}$ close to a peak in the density of states (dashed line). This value is close to the experimental observations ($\approx -0.5\times {10}^{12}{\mathrm{cm}}^{-2}$) (11). We find that the Fermi-surface term (blue) arising from ${\alpha }_{n\mathit{k}}$ dominates ${\chi }_{\mathit{me}}$; the Fermi-sea contribution (red) is minor. Additionally, we plot ${\chi }_{\mathit{me}}$ as a function of ${\mathrm{\Delta }}_d$ in Fig. 2D. Upon increasing the layer potential ${\mathrm{\Delta }}_d$, ${\chi }_{\mathit{me}}$ generally decreases due to the weakened orbital magnetoelectric moment. This indicates that a large LMC response is more likely to occur in the regime of small ${\mathrm{\Delta }}_d$.

Butterfly-Shaped Hysteresis.

As we now explain, LMC directly couples with valley order to produce an electric field switching of magnetic order recently seen in the experiment of ref. 11. In doing so, we formulate a phenomenological Landau theory for the valley-polarized order with LMC. To understand the interplay between the electric field and valley polarization, we first adopt a small but fixed B field that contributes a magnetoelectric like contribution to the free-energy density: $-\delta MB-\delta PE=-({\chi }_{\mathit{me}}+{\chi }_{\mathit{em}})EB=-\mathrm{\Gamma }{\mathrm{\Delta }}_{v}E$. Such fixed B fields were present in the switching devices of ref. 11. Γ parameterizes an emergent valley-electric coupling parameter $\mathrm{\Gamma }=2{\chi }_{\mathit{me}}B$ mediated by the orbital LMC. As a result, the Landau free energy for the valley-polarized order can be written as

$$\begin{array}{c}\hfill f_L=-a{\mathrm{\Delta }}_v^2+\frac{b}{2}{\mathrm{\Delta }}_v^4-\mathrm{\Gamma }{\mathrm{\Delta }}_{v}E,\end{array}$$ [15]

where $a,b$ are phenomenological Landau parameters describing the ordering transition. In what follows, we will treat $a,b,\mathrm{\Gamma }$ phenomenologically to demonstrate the coercive-hysteretic behavior.

Below the critical temperature, we have $a>0$, $b>0$, and the valley-polarized order ${\mathrm{\Delta }}_v=\pm \sqrt{a/b}$. As shown in Fig. 3A, the free energy landscape as a function of ${\mathrm{\Delta }}_v$ evolves under different electric field strengths. Starting from a state with ${\mathrm{\Delta }}_v<0$, the order parameter evolves as an electric field is applied. When the electric field reaches coercive field ($E=E_c$) (see orange curve), the local minimum at ${\mathrm{\Delta }}_v<0$ of the free energy collapses (it becomes unstable) and transitions to a global minimum at ${\mathrm{\Delta }}_v>0$. This coercive field can be derived as

$$\begin{array}{c}\hfill E_c=\frac{4a}{3\mathrm{\Gamma }}\sqrt{\frac{a}{3b}}\end{array}$$ [16]

Fig. 3.

(A) The free energy landscapes at different electric fields. (B) The magnetization butterfly-shaped hysteresis curve induced by the electric field. The coercive field is denoted as $E_c$. (A and B) are evaluated phenomenologically from Eq. 15.

Interestingly, since $E_c$ becomes large for small Γ, the coercive fields rapidly increase for small B rendering electric field induced switching challenging. This behavior is consistent with reports of increasing coercive fields with decreasing magnetic field (11); nevertheless modest values $B\sim 0.1 T$ readily produce accessible $E_c$ as seen in ref. 11.

The coercive field induced transitions directly produce butterfly-shaped hysteretic curves as a function of E. To see this, we plot the magnetization $\delta M={\chi }_{\mathit{me}}E$ as a function of E in Fig. 3B. Here, magenta (blue) curves denote valley order with ${\mathrm{\Delta }}_v<0$ (${\mathrm{\Delta }}_v>0$). Because ${\chi }_{\mathit{me}}$ is locked to ${\mathrm{\Delta }}_v$, the diagonal slopes of the magnetization vs. E curves directly reflect the valley order. In contrast, the vertical lines track the transitions between ${\mathrm{\Delta }}_v<0$ and ${\mathrm{\Delta }}_v>0$ ordered states. Note the butterfly pattern flips when B changes sign (i.e. $\mathrm{\Gamma }⟷-\mathrm{\Gamma }$) mirroring the flipping of the hysteretic pattern seen in ref. 11. Such a butterfly has also been observed in numerical calculations (33).

Nonlinear Magnetoelectric Coupling.

While the linear orbital LMC above requires both broken time reversal and broken inversion symmetry, nonlinear LMC, on the other hand, have less stringent symmetry requirements. For example, ${\gamma }_e$ (Eq. 1) only requires broken time-reversal and ${\gamma }_b$ (Eq. 2) only requires broken inversion symmetry. As a demonstration we evaluate ${\gamma }_b$ for time-reversal symmetric dual-gated graphene multilayers.

Using a Green’s function approach to evaluate B field induced changes (46, 47) to the free energy, we obtain the nonlinear polarization (induced by $B^2$) coefficient ${\gamma }_b$ as

$$\begin{array}{c}\hfill {\gamma }_b=\frac{e^3{ħ}^2}{4\beta }{\displaystyle \sum _{n\mathit{k}}}\mathrm{Tr}\{\widehat{\mathit{p}}G[{({\widehat{v}}_{x}G{\widehat{v}}_{y}G)}^2+{({\widehat{v}}_{y}G{\widehat{v}}_{x}G)}^2]\},\end{array}$$ [17]

where $G={[i{\omega }_n+E_F-H_0(\mathit{k})]}^{-1}$ is the single-particle Matsubara Green’s function with ${\omega }_n=2(n+1)\pi /\beta$. Here, ${\gamma }_b$ is expressed in compact Green’s functions form without expanding into multiple terms (47, 48) making it straightforward for numerical evaluation. It is worth noting that the Fukuyama formula (46) was derived for Bloch electrons in the continuum limit, whereas additional modifications can arise in tight-binding models (49).

As an illustration, we evaluate ${\gamma }_b$ for the time-reversal symmetric phase of multilayer graphene in Fig. 4 (RPG in B and rhombohedral trilayer graphene in D). While multilayers possess pronounced ${\gamma }_b$ in its metallic phase (green), most striking is the presence of a nonvanishing plateau in the insulating phase (yellow). Such a nonlinear LMC effect captures a Fermi-sea contribution and produces sizeable nonlinear polarizations. For instance, at modest magnetic field $B=2$T, we find the induced polarization $\delta P\sim 4\mathrm{n}C/c{m}^{-2}$ in the insulating regime and can reach $30\mathrm{n}C/c{m}^{-2}$ in the metallic regime. Interestingly, such magnetic field induced nonlinear polarizations have values comparable to nonlinear polarizations induced at second-order in the electric field (36, 37, 50) underscoring the strength of the LMC in multilayer graphene.

Fig. 4.

(A) The band structure of RPG with the layer potential ${\mathrm{\Delta }}_d=20$ meV. (B) The nonlinear LMC coefficient ${\gamma }_b$ as a function of Fermi energy $E_F$. The insulating and metallic regions are labeled by yellow and green colors, respectively. (C and D) The results for RTG with the layer potential ${\mathrm{\Delta }}_d=20$ meV. The temperature is set to be $T=5$K.

Conclusion and Discussion.

In this work, we have presented a compact microscopic theory of LMC in multilayer graphene and elucidated its orbital (and layer degree of freedom) origin. Notably, we identify an orbital magnetoelectric moment (Eq. 10) as a key quantum geometric quantity (in layer space) that dominates the magnetoelectric response in multilayers. We note that the generation of nonlinear magnetization (51) and layer electric polarization has recently garnered intense recent interest (36, 37, 50). Our general result for nonlinear LMC highlights opportunities for controlling layer polarization, applicable to a broad range of noncentrosymmetric layered materials (52) and can be experimentally observed via capacitance probes (53, 54).

Materials and Methods

The orbital magnetoelectric coupling in layered systems, as discussed in the main text, was examined using a perturbative analysis; hysteresis was analyzed using a phenomenological Landau theory with a layer orbital magnetoelectric term. Additional theoretical details of the results presented in the main text are also provided in SI Appendix. SI Appendix is organized as follows: SI Appendix, section I details the microscopic theory of longitudinal magnetoelectric coupling. It includes the definition of the layer polarization operator, layer-space quantum geometry, derivation of magnetoelectric coefficients, the Hellmann–Feynman theorem, and discussion of the effect of screening. SI Appendix, section II provides a phenomenological theory for the butterfly-shaped hysteresis. SI Appendix, section III presents a derivation of the nonlinear magnetoelectric coupling in layered systems. SI Appendix, section IV discusses the longitudinal magnetoelectric effect in layered antiferromagnetic thin films.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix.