Designer meron lattice on the surface of a topological insulator

We present a promising route to realize spontaneous magnetic order on the surface of a 3D topological insulator by applying a superlattice potential. The superlattice potential lowers the symmetry of the surface states and creates tunable van Hove singularities, which, when combined with strong spin-orbit coupling and Coulomb repulsion, give rise to a topological meron lattice spin texture. The periodicity of this designer meron lattice can be tuned by varying the periodicity of the superlattice potential. We employ Ginzburg-Landau theory to classify the different magnetic orders and show that the magnetic transition temperature reaches experimentally accessible values. Our work introduces another direction to realize exotic quantum order by engineering interacting Dirac electrons in a superlattice potential, with promising applications to spintronics.

Figure 1

Schematic of experiment and meron spin texture. (a) The experiment consists, from top to bottom, of a metallic gate, a patterned dielectric, and a topological insulator. Applying a bias between the metallic gate and the topological insulator imposes a modulated potential on the surface of the TI. (b) Magnetization exhibiting half-integer winding characteristic of a meron spin texture. Merons pinned by the superlattice potential form a meron lattice.

Figure 2

Superlattice Brillouin zone, electronic properties of surface states, and magnetic ordering basis. (a) Black and cyan lines show the superlattice and magnetic Brillouin zones, respectively. The red and green arrows represent the wave vectors ${\mathbf{q}}_j$ and the nesting vectors ${\mathit{\delta }}_j$, respectively. (b) Density of states and electronic dispersion for $w/(v_F/L)\simeq 1.54$ ($L=20\mathrm{nm}$, $w=20\mathrm{meV}$). Dashed lines show the density of states and the band structure of the TI Dirac cone folded into the superlattice Brillouin zone. (c) Momentum-resolved spectral function displaying the Fermi surface at the energy of the VHS with $w/(v_F/L)\simeq 1.54$. (d) Decomposition of the order parameter into ${\mathit{v}}_{\mu j}$.

Figure 3

Saddle-point solution, real-space magnetization, and electronic structure in the meron lattice phase. (a) Blue and red data show the total magnetization $\left|M\right|$ and the lowest eigenvalue of ${\mathcal{M}}_{jl}^{\mu \nu }$, respectively. (b) Contour plot of the magnetization in real space. Color indicates the out-of-plane component. (c) Winding number $\mathit{m}·({\partial }_x\mathit{m}\times {\partial }_y\mathit{m})/4\pi$ in real space. Black and cyan lines show the superlattice and magnetic unit cells, respectively. (d) Density of states and electronic dispersion of the meron lattice state. The dashed line shows the density of states of the normal phase. The horizontal line shows the Fermi energy $E_F$; $\mathrm{\Delta }E$ denotes the induced gap at ${\mathit{\Gamma }}_m$. Inset shows the Fermi surface of the magnetic state. The bands are obtained from the Hartree-Fock solution at $w/(v_F/L)\simeq 1.54$, $2w=40\text{meV}$, $U=30\text{meV}$, and $T=1.5\text{K}$.

Figure 4

Magnetic instability, critical temperature, and interaction. (a) Solid lines show the components of the second-order free energy $L_{\mu \nu }$. Red and cyan dashed lines illustrate the eigenvalues ${\lambda }_{-,2}$ and ${\lambda }_{+,2}$, respectively, while ${\lambda }_1=L_{\perp \perp }$ is the solid green line. The calculations were performed at $T=1.5\text{K}$, $U=30\text{meV}$, and $w/(v_F/L)\simeq 1.54$. (b) Critical temperature for the magnetic transition as a function of $w/(v_F/L)$ for $U=30\text{meV}$. For a given $w/(v_F/L)$, the filling $n$ is set so that the Fermi level is at the energy of the VHS. (c) Evolution of the energy of the VHS singularity as a function of $w/(v_F/L)$. Inset shows the electron filling per superlattice unit cell at the VHS versus $w/(v_F/L)$. (d) Critical interaction $U_c$ as a function of the filling $n$ at several temperatures, fixing $2w=40\text{meV}$ and $w/(v_F/L)\simeq 1.54$.

Figure 5

Parameters $\alpha$ and $\eta$ of the $\mathbf{k}·\mathbf{p}$ expansion of the dispersion relation around $\mathbf{K}$. The vanishing of $\alpha$ at $w/(v_F/L)=1.36$ highlighted by the green line implies a high-order van Hove singularity.