Hopper Growth of Higher-Order Topological Insulators

TL;DR

The paper demonstrates that intrinsic topological properties of higher-order topological insulators, specifically obstructed atomic insulators, can drive hopper-shaped crystal growth under diffusion-limited conditions. By connecting corner states to deposition energetics through a microscopic framework based on Wannier centers and charging arguments, it shows that topological phases favor rapid corner advancement and produce distinctive hopper morphologies, distinct from dendritic growth at the same overall fractal dimension $D_f$. The authors quantify morphology using fractal dimensions $D_f$ and $D_{f,c}$, revealing that for fixed $D_f$ the topological phase exhibits a smaller $D_{f,c}$, corresponding to smoother, stepwise-depressed hopper boundaries due to topological state contributions. The work unifies topological electronic structure with crystal growth via energy-based deposition rules, suggests candidate obstructed-insulator materials, and lays groundwork for extending the framework to three dimensions.

Abstract

Understanding crystal growth and morphology is a fundamental issue in condensed matter physics. While crystal morphology due to the distribution and dynamics of the diffusion field has been intensively studied, how the intrinsic material properties affect crystal morphology remains unclear. In this Letter, we demonstrate that higher-order topological phases can give rise to hopper-shaped crystal morphologies through an unconventional mechanism originating from topological electronic states. We quantitatively show this connection by analyzing both the fractal dimension $D_f$ and the fractal dimension of coastlines $D_{f,c}$. When we compare the crystals in the trivial and topological phases with the same $D_{f}$ in the case of relatively rapid crystal growth, the former is in the dendritic shape, while the latter is in the hopper shape, quantified by the smaller $D_{f,c}$ in the topological phase. We reveal the microscopic origin of this anomalous growth in the higher-order topological phase, and find that it leads to the stepwise-depressed morphology characteristic of hopper crystals. Our theory uncovers a fundamental link between hopper crystals and higher-order topological phases, offering unconventional insight into crystal morphology governed by topological electronic states.

Figures (8)

• Figure 1: (a) The positions of ions and Wannier centers for the 2D model $\mathcal{H}(\boldsymbol{k})$. (b) A crystal with electronic orbitals located near the corners of the unit cell, which can be regarded as a covalent crystal between orbitals in the neighboring atoms. Black circles are the electronic orbitals, and the lines between the electrons indicate the covalent bonds. (c) A square-shaped crystal with an atom deposited at the corner. (d) A square-shaped crystal with an atom deposited next to the corner. (e-g) The number of dangling bonds (green segments) changes at the deposition of an atom onto the site at the dotted circle.
• Figure 2: (a,b) Crystal shapes obtained from the simulations. The blue, light blue, and light green sites indicate the solid atoms, interfacial solid atoms, and gas atoms, respectively. The parameters in Eq. (\ref{['eq:energy_change_approx']}) are set to (i) $e_b=0.6439+0.2413+0.1488$, $e_s = 0$, $e_t = 0$ in (a) corresponding to the trivial phase and (ii) $e_b=0.6439$, $e_s = 0.2413$, $e_t = 0.1488$ in (b) corresponding to the nontrivial phase, which are the values obtained by the fitting to the numerical result [Fig. \ref{['fig:energy_change']}] in End Matter. The crystal shapes are obtained with $t=1500$ and $\Delta \mu/k_{\rm B}T = -8.75$. We choose $k_{\rm B}T=0.16$ in both the (i) and (ii) cases. (c) Fractal dimensions $D_f$. (d) The fractal dimension of coastlines $D_{f,c}$. Here, "trivial" and "nontrivial" indicate the parameters (i) and (ii), respectively. The error bars indicate the standard error of the mean of six simulations for each value of $\Delta \mu/k_{\rm B}T$. To reduce statistical uncertainty, we increase the number of Monte Carlo samples to twenty at $|\Delta \mu|/k_BT \geq 10$ for the parameters (i) and at $|\Delta \mu|/k_BT \geq 9$ for the parameters (ii). The fits for the fractal dimensions are performed only for data points with $N_{s,i}>100$.
• Figure 3: (a) Fractal dimensions $D_f$ versus fractal dimensions of coastlines $D_{f,c}$. Here, "trivial" and "nontrivial" indicate the parameters (i) and (ii) in Fig. \ref{['fig:growth_shape']}, respectively. The error bars indicate the same standard errors as those in Fig. \ref{['fig:growth_shape']}. (b) The relationship between the logarithm of $N_s$ and the logarithm of $N_{s,i}$, where the logarithm base is 10. The fits are performed only for data points with $N_{s,i}>100$. (c,d) The obtained crystal shapes for (c) the nontrivial case with $\Delta \mu/k_{\rm B}T=-9.2$ and (d) the trivial case with $\Delta \mu/k_{\rm B}T=-11.3$, with $8000\leq N_{s} \leq 8010$. (c-1) and (d-1) correspond to the results in (b).
• Figure 4: Representative examples of the growth process that increase the number of atoms at the interface $N_{s,i}$.
• Figure 5: (a-c) The energy changes upon the deposition of an atom onto crystals with (a) $N$ atoms (b) $N+1$ atoms, and (c) $N+2$ atoms. The positions of the deposition are labeled with $i$ in (a), $j$ in (b), and $k$ in (c). TB and ET indicate the calculations from the tight-binding model $\mathcal{H}(\boldsymbol{k})$ and those from the effective theory given in Eq. (\ref{['eq:energy_change_approx']}), respectively. We choose the parameters $t=1$ and $m=v=\Delta = 0.5$ with the system with a square shape of $30\times 30$. The parameters of the approximation in Eq. (\ref{['eq:energy_change_approx']}) are set to $\mu_s = -4.984$, $e_b=0.6439$, $e_s = 0.2413$, and $e_t = 0.1488$.