Current switching behaviour mediated via hinge modes in higher order topological phase using altermagnets

TL;DR

This work shows that proximitizing a 3D topological insulator with $d$-wave altermagnets yields both hybrid-order and second-order topological phases, tunable via the relative AM strengths. A HyOTP combines gapless surface Dirac states with hinge modes, diagnosed by dipolar and quadrupolar winding numbers and corroborated by transport signatures; and, when both $d$-wave AM components are present, two distinct SOTIs with hinge-mode directionality arise, enabling a hinge-mediated current-switching mechanism. The authors develop low-energy surface theories, derive effective boundary Hamiltonians, and demonstrate hinge/domain-wall physics through Jackiw-Rebbi-type modes, while also outlining a speculative TOTI route with an octupolar invariant and corner states. These insights position altermagnet-based hybrids as versatile platforms for controllable higher-order topology and novel hinge/edge-device functionalities, with potential material realizations via TI–RuO$_2$ heterostructures.

Abstract

We propose a theoretical framework to engineer hybrid-order and higher-order topological phases in three-dimensional topological insulators by coupling to $d$-wave altermagnets (AMs). Presence of only $d_{x^2-y^2}$-type AM drives the system into a hybrid-order topological phase where both first-order and second-order topological phases coexist. This phase is characterized by spectral analysis, low-energy surface theory, dipolar and quadrupolar winding numbers, and it's signature is further confirmed by two-terminal differential conductance calculations. Incorporation of the $d_{x^2-z^2}$-type AM drives the system into two second-order topological insulator phases hosting distinct type of hinge modes. They are also topologically characterized by spectral analysis, topological invariants, low-energy surface thoery, and transport calculations. Importantly, the localization and direction of propagation of these one-dimensional hinge modes are controllable by tuning the relative strengths of the alermagnetic exchange orders. We utilize this feature to propose a tunable current-switching behaviour mediated via the hinge modes. Our results establish AMs based hybrid structure as a versatile platform for controllable higher-order topology and hinge-mediated device applications.

Figures (7)

• Figure 1: Schematic diagram of our setup: Schematic illustration of a 3D first-order topological insulator (purple, top) placed on the surface of a $d$-wave AM (orange, bottom) with both $(x^2-y^2)$ and $(x^2-z^2)$ (not shown) symmetry components. The 1D hinge states are shown by the yellow lines propagating along $\pm z$ directions (marked by the red arrows). The top surface hosts gapless Dirac cones, while the front surface is gapped.
• Figure 2: Spectral properties and topological invariants in the HyOTP: In panel (a), we show the LDOS ($E=0$) in the HyOTP considering a finite geometry along the $x,y,$ and $z$ directions with $L_x=L_y=L_z=12$ lattice sites. Panel (b) displays the energy spectrum, corresponding to finite system size along $z$ directions with $L_{z}=100$, as a function of $k_y$ with $k_x=0$, while the inset exhibits the spatial profile $|\psi(z)|^2$ at $k_x=k_y=0$. Panel (c) shows the hinge spectrum as a function of $k_z$, with insets ($I_1$, $I_2$) illustrating the LDOS at $E=0$ and the eigenvalue spectrum with respect to eigenstate index $n$ with $L_x=L_y=20$. In panel (d), we present the topological invariants, DWN $W$ (left axis) and QWN $N_{xy}$ (right axis) as a function of $m_0$. Other model parameters ae chosen as, $m_0 = t$, $t=1$, $\lambda=t$, and $J_{xy}=2t$.
• Figure 3: Transport characteristics in the HyOTP: In panel (a), we depict the differential conductance $\frac{dI^{\rm (S)}}{dV}$ as a function of $(eV)$, due to the surface states in HyOTP. The inset illustrates the transport setup with LL (left lead), RL (right lead), and S (system) through which the current flows. Panel (b) presents the differential conductance $\frac{dI^{\rm (H)}}{dV}$ with respect to $(eV)$ for the hinge states. The inset (bottom) highlights the transport setup with top and bottom leads (TL and BL). On the other hand, the inset (top) emphasizes the $4e^{2}/h$ contribution arising due to hinge modes of the main plot. For panel (a) $L_{z}=100$ and panel (b) $L_{x}=L_{y}=20$. Other model parameters remain same as mentioned in Fig. \ref{['Fig2']}.
• Figure 4: Topological characterization and current switching behavior in the SOTI phase. We depict $N_{xy}$ and $N_{xz}$ for the $xy$- and $xz$-planes in panels (a) and (b), corresponding to SOTI$^{\text{(I)}}$ and SOTI$^{\text{(II)}}$ phases respectively. Insets (I1) and (I2) illustrate the eigenvalue spectrum $E_{n}$ with respect to $n$. The LDOS distribution at $E=0$ for both the cases are also shown in panels (a) and (b). Panel (c): $dI^{\rm (H)}/dV$ is displayed as a function of bias $eV$ for the SOTI$^{\rm (I)}$ ($J_{xz}=2t$) and SOTI$^{\rm (II)}$ phase ($J_{xz}=4t$); the inset exhibits an expanded view of the same conductance plot to highlight the contribution arising from the hinge modes. (d) $dI^{\rm (H)}/dV(eV=0)$ is illustrated as a function of $J_{xz}$, with the corresponding transport device setup given in the inset. The other model parameters are chosen as, $m_0 = t$, $t=1$, $\lambda=t$, $J_{xy}=3t$ and $L_{x}=L_{y}=20$.
• Figure S1: 2D Surface states and corresponding LDOS for different values of $J_{xy}$ are shown. Panels (a–c) exhibit the surface spectra for $J_{xy}=0$, with the corresponding LDOS at $E=0$ presented in panel (d). Panels (e–g) display the surface spectra for $J_{xy}=2t$, and the corresponding LDOS is shown in panel (h). Other model parameters are set to be $m_0=t$, $\lambda=t$ and $t=1$.