Sierpiński Structure and Electronic Topology in Bi Thin Films on InSb(111)B Surfaces

Deposition of Bi on InSb(111)B reveals a striking Sierpiński-triangle (ST)-like structure in Bi thin films. Such a fractal geometric topology is further shown to turn off the intrinsic electronic topology in a thin film. Relaxation of a huge misfit strain of about 30% to 40% between Bi adlayer and substrate is revealed to drive the ST-like island formation. A Frenkel-Kontrova model is developed to illustrate the enhanced strain relief in the ST islands offsetting the additional step energy cost. Besides a sufficiently large tensile strain, forming ST-like structures also requires larger adlayer-substrate and intra-adlayer elastic stiffnesses, and weaker intra-adlayer interatomic interactions.

Figure 1

(a),(b) STM images of typical ST-like Bi islands formed on InSb(111)B substrate at $\theta \approx 2\mathrm{ML}$. See Sec. S1 for details. (c) Schematic for different island height levels corresponding to the labels in (a),(b). (d),(e),(f) Height profiles for red, blue, and pink lines $A$, $B$, and $C$ in (b), respectively.

Figure 2

(a) STM image at $\theta \approx 1\mathrm{ML}$. See Supplemental Material, Sec. S1 [44] for details. (b) Height profile for a red line in (a). (c) The same image as (a). Side lengths of STs are indicated. (d) Atomistic structure model for a 1-ML Bi ST with all side lengths accurately matching the lengths measured in (c). A purple ball denotes a lattice site occupied by a Bi atom and a gray ball denotes an unoccupied lattice site.

Figure 3

Shapes and energies of four configurations after energy minimization for 1-ML Bi islands with the same number of Bi atoms, $N=267$. A purple ball represents a Bi atom and a light ball represents a lattice point below this Bi atom. For $C0$, $C1$, and $C2$, $b=b_{1\mathrm{ML}}=0.3304\mathrm{nm}$, $k_2=k_{1\mathrm{ML}}=766\mathrm{eV}/{\mathrm{nm}}^2$, and $\varphi ={\varphi }_{1\mathrm{ML}}=-0.747\mathrm{eV}$; for $C3$, $b=b_{\mathrm{chain}}=0.2865\mathrm{nm}$, $k_2=k_{\mathrm{chain}}=814\mathrm{eV}/{\mathrm{nm}}^2$, and $\varphi ={\varphi }_{\mathrm{chain}}=-1.638\mathrm{eV}$; for all, $l=a_0=0.45816\mathrm{nm}$ and $k_1=k_{\mathrm{sub}}=307\mathrm{eV}/{\mathrm{nm}}^2$.

Figure 4

DFT chemical potentials $\mu$ (the energy per Bi atom relative to gas phase) versus lateral lattice parameter $b$ for freestanding (a) 1-, (b) 2-, (c) 3-, and (4) 4-ML Bi slabs. Interlayer spacings ($d_1$, $d_2$, and $d_3$) for 2-, 3-, and 4-ML slabs, as well as magnetic moments $m$ for 1 ML in (a) and chain in the inset of (a), are also plotted. For details, see Sec. S3.2.