Shape dependence of Edelstein and magnetoelectric effects in the V-shaped model

TL;DR

This paper addresses how a V-shaped geometry in a 1D chain controls magnetoelectric responses and unveils the underlying microscopic mechanisms. By combining a full $s$–$p$ orbital model with a low-energy projection to an effective $s$-orbital Hamiltonian, the authors reveal that the geometry-induced polarity acts as an emergent spin–orbit interaction and that the shape effect is encoded in a $T$-matrix attached to the apex. A multipole-basis analysis identifies selection rules showing that coupling between the effective SOC and orbital angular momentum generated across the apex governs the ME response, producing an angular dependence $\sin\theta \sin(\tfrac{\theta}{2})$ with a peak at $θ = 2\tan^{-1}\sqrt{2} \approx 0.608π$. In the ferromagnetic version, Zeeman coupling yields a magnetic-driven ME effect sourced by field-induced spin magnetization, yielding different angular and size dependences. Together, these results establish a microscopic, geometry-driven framework for shape-induced multipole phenomena in both mesoscopic and bulk systems, with potential relevance to engineered metamaterials and zigzag chains.

Abstract

We theoretically investigate the shape dependence and microscopic mechanism of the magnetoelectric effect, including both nonmagnetic (Edelstein-type) and magnetic origins, in a V-shaped one-dimensional chain model. Numerical calculations based on the Kubo formula reveal that the magnitude of the nonmagnetic-driven magnetoelectric response reaches a maximum at an apex angle of $θ\approx 0.6π$. To clarify the microscopic origin of this behavior, we construct a low-energy effective Hamiltonian by projecting onto the $s$-orbital subspace and demonstrate that the polarity induced by the V-shaped geometry manifests as an effective spin--orbit interaction. An analytical derivation of the Green's function shows that the geometric effect appears as a $T$-matrix contribution, reflecting the local breaking of translational symmetry at the V-shaped edge. Furthermore, by employing a multipole-basis representation, we identify the selection rules that govern the magnetoelectric tensor and reveal that the coupling between the effective spin--orbit interaction and the orbital angular momentum generated across the apex plays an essential role. The resulting angular dependence, $\sinθ\sin{θ/2}$, peaks at $θ= 2\tan^{-1}\sqrt{2} \approx 0.608π$, in good agreement with the numerical results. We also analyze a ferromagnetic V-shaped model including the Zeeman interaction and show that the magnetic-driven magnetoelectric response originates from the spin magnetization induced by the coupling between the electric-field--driven charge-potential gradient and the Zeeman term. These findings demonstrate that the V-shaped geometry gives rise to distinct magnetoelectric mechanisms depending on the presence or absence of time-reversal symmetry and provide a microscopic framework for understanding shape-induced multipole phenomena in mesoscopic and bulk systems.

Figures (8)

• Figure 1: The lattice of the $2N+1$ -site $(N=4)$ V-shaped 1D chain defined in Eq. (\ref{['eq:kulat']}). $a$ is the lattice constant and $\theta$ is the apex angle.
• Figure 2: (a) Chemical potential $\mu$ dependence of the nonmagnetic-driven ME tensor component $\alpha^{z;x}$ for various apex angles, $\theta=0.0\pi,0.2\pi,\ldots,1.0\pi$. $\alpha^{z;x}$ reaches its maximum around $\mu \approx 1.5$ and its minimum around $\mu \approx -1.45$, independent of the value of $\theta$. (b) Apex angle $\theta$ dependence of the ME tensor at $\mu=0.5$. The magnitude of the ME response induced by the V-shaped geometry reaches its maximum at $\theta \approx 0.6\pi$. In both (a) and (b), the parameters are set to $t_{ss\sigma}=1, t_{sp\sigma}=0.7, \Delta_{p}=10, \lambda=0.4, N=128, \gamma=0.05$, and $T=0.01$.
• Figure 3: (a) Diagrams representing the effective Green's function in Eq. (\ref{['eq:effgre']}). The thick black line represents the effective Green's function $\bar{\mathcal{G}}_{\mathrm{eff}}$ in Eq. (\ref{['eq:effgre']}), the thin black line denotes the Green's function propagating between $s$ orbitals $\bar{g}_{\parallel}$, and the gray double line represents the Green's function propagating between $p$ orbitals $\bar{g}_{\perp}$. The cross symbol ($\times$) indicates the $s$--$p$ orbital hybridization $\bar{\eta}$. As shown in Eq. (\ref{['eq:geffhyb']}), the effective Green's function represents the result of incorporating the $p$-orbital effect into the $s$-orbital Green's function through the $s$--$p$ hybridization. (b) Diagrams representing the effective magnetization operator in Eq. (\ref{['eq:effMag']}). The vertex accompanied by a wavy line represents the magnetization operator $\bar{M}^{\nu}$. (c) Diagrams representing the effective ME tensor in Eq. (\ref{['eq:effsus']}). The $p$-orbital contribution is included via the $s$--$p$ hybridization in the bubble diagram closed within the $s$-orbitals.
• Figure 4: Absolute values of the components of (a) $\bar{\mathcal{T}}^{(\theta)}$ and (b) $\Delta \bar{M}^{z(\theta)}_{\mathrm{eff}}$ as a function of the apex angle $\theta$. We fix the chemical potential $\mu=0.5$ and the complex frequencies $z=z_1=z_2=0.5\pi i$, and we use the other model parameters in Eq. (\ref{['eq:HamPara']}). In both cases, the terms including $q_0^{\mathrm{c}}$ and $q_0^{\mathrm{b}}$ exhibit a $(1+\cos{\theta})$-type behavior, whereas that including $t_x^{\mathrm{b}}$ shows a $\sin{\theta}$-type dependence. Furthermore, the obtained results display the $\Delta_{p}$ dependence consistent with Eqs. (\ref{['eq:effMag']}) and (\ref{['eq:Tthdep']}).
• Figure 5: Diagrams representing distinct processes for the ME effect in our effective models. The thick black line represents the simple 1D chain Green's function at $\theta=\pi$, and the circled T (Ⓣ) represents the $T$-matrix about the V shape in Eq. (\ref{['eq:tmatrix']}). The subscripts of the Matsubara frequencies are omitted. The symbols (e.g., $P_{q_x^{\mathrm{c}}\sigma_0}$) attached to the operators correspond to their multipole components in Table \ref{['tab:MpBofOperator']}. The physical mechanisms corresponding to each multipole component are shown along with their schematic illustrations (see also Fig. \ref{['fig:MpB']} in Appendix \ref{['sec:app:MpB']}). There are two primary physical mechanisms contributing to the ME effect: One is the coupling between the charge-potential gradient induced by the incident electric field and the orbital magnetization ($\alpha^{z;x}_{\mathrm{i}}$ and $\alpha^{z;x}_{\mathrm{ii}}$). The other is the spin magnetizations induced by the electric field through the effective spin--orbit interaction ($\alpha^{z;x}_{\mathrm{iii}}$ and $\alpha^{z;x}_{\mathrm{iv}}$).