Origin of the moiré pattern in thin Bi films deposited on HOPG

Thin Bi(110) films deposited on highly oriented pyrolytic graphite (HOPG) exhibit a pronounced moiré pattern; here the origin of the moiré pattern is investigated using scanning tunneling microscopy (STM) and spectroscopy (STS), high-resolution transmission electron microscopy (HR-TEM), and density functional theory (DFT). It is shown that the moiré pattern forms only on islands which are misoriented by $\sim {30}^{\circ }$ with respect to the usual substrate symmetry direction. Two models of the moiré pattern are presented: (i) a commensurate monolayer construction (CMC) for rectangular overlayer symmetry on hexagonal substrates and (ii) a qualitative model based on simple superposition of a Bi overlayer on graphene. The CMC model has previously been applied only to systems with pure hexagonal symmetry. Both models generate moiré patterns with key parameters (period, angles of the pattern measured with respect to the main HOPG and Bi crystal directions) that are consistent with the experimental results, but development of a fully predictive/quantitative model remains an outstanding challenge. The electronic structure of the moiré pattern is investigated using STS and DFT, and it is found that the local density of states (LDOS) is modulated by the moiré pattern. These results are consistent with a picture in which a small distortion of Bi atomic positions at the film-substrate interface results in periodic modulation of the LDOS, hence allowing observation of the moiré pattern in STM images.

Figure 1

Bi(110) crystallographic structure and model grain boundary. (a) Intersection of two grains at a tilt grain boundary showing the unit cell dimensions $A,B$, interrow spacing $d$, and key crystallographic directions in each grain. See Appendix pp1 for details of the description of the grain boundary. Cross-section views (along the $\langle \overline{1}10\rangle$ direction) for (b) bulk-like and (c) black-phosphorus-like structures.

Figure 2

(a) High-resolution STM image ($V=-2\mathrm{V}$, $I=1.5$ nA) recorded at 300 K clearly showing the moiré pattern ($\lambda =4.0$ nm) recorded in the rectangular region of the low-resolution image in the inset. (b) Distribution of measured moiré angles $\delta$. (c) Distribution of measured moiré periodicities $\lambda$. Black, red, and blue markers in (b) and (c) correspond to $\delta$ and $\lambda$ obtained from the CMC model for three sets of indices $(n,m,u,v)$; see text.

Figure 3

(a) STM image ($V=-0.8\mathrm{V}$, $I=10\text{pA}$) of a 3-ML-thick Bi island recorded at 50 K showing two grains (labeled 1 and 2) separated by a $\Sigma 1$ tilt grain boundary. The inset in (a) shows a fast Fourier transformation enhanced STM image ($V=-0.3\mathrm{V}$, $I=10\text{pA}$) of the rectangular region marked in the main plot, showing that the moiré pattern ($\lambda =4.4$ nm) is observed only in grain 1. (b) Magnification of the central part of the inset showing atomic resolution ($V=-0.3\mathrm{V}$, $I=0.5$ nA). Unit cells in grains 1 and 2 are $0.44\times 0.48$ and $0.46\times 0.48{\text{nm}}^2$, respectively. (c) Model of $\Sigma 1$ tilt grain boundary. $\mathrm{Bi}\langle \overline{1}10\rangle$ is nearly parallel to the HOPG armchair ($\langle 10\overline{1}0\rangle$) direction in grain 1 and is parallel to the HOPG zigzag ($\langle 1\overline{1}00\rangle$) direction in grain 2. Armchairs and zigzags for HOPG are indicated using thick lines in (c). (d) Left: Bi surface unit cell for grain 1, center: HOPG unit cell, and right: Bi surface unit cell for grain 2, showing the main crystallographic directions.

Figure 4

(a) HR-TEM image recorded on an island with a $\Sigma 1$ tilt grain boundary (marked with blue arrows). Grains 1 and 2 are bordered using red and green lines, respectively. The image was enhanced using FFT filtering in order to more clearly show the moiré pattern in grain 1. (b) FFT of the image in (a). Reciprocal unit cells for grains 1 (red) and 2 (green) are shown. FFTs calculated for (c) grain 1 and (d) grain 2 individually. Clear moiré spots (marked M) are seen in (c), corresponding to $\lambda \sim 3.5\text{nm}$. Crystallographic directions of $\mathrm{Bi}\langle \overline{1}10\rangle$, $\mathrm{HOPG}\langle 10\overline{1}0\rangle$, and $\theta$, $d$, $\gamma$ are shown in (c) and (d).

Figure 5

(a) STM image ($V=1.0\mathrm{V}$, $I=1.0$ nA) showing a 3-ML-thick region with a 5-ML stripe. (b) $dI/dV$ image for $V=+100$ mV corresponding to the topographic image in (a), revealing the moiré pattern in the 3-ML region. (c) FFT power spectrum calculated for (b). Red circles indicates spots corresponding to the moiré pattern in (b). (d) Tunneling conductance map recorded along the line shown in (a) and (b). (e) Two tunneling conductance curves recorded on moiré intensity minimum and moiré intensity maximum as indicated by arrows 1 and 2 in (b). These positions are also indicated using vertical lines in (d). Line 3 is the normalized FFT intensities (see Appendix pp2) as a function of the energy recorded for moiré pattern [red circles in (c)].

Figure 6

Results of calculations using commensurate monolayer construction. (a) Solutions ($a_{rh}$, $\theta$) of Eqs. (5) and (6) for $\alpha =57.3^{\circ }$. (b) Solutions for $\alpha =54.8^{\circ }$. The color scale is an indication of the distance between adjacent commensurate sites, $z=\sqrt{n^2+m^2}<10$. Arrows highlight indices $(n,m,u,v)$ discussed in the text. (c) Dependence of $a_{rh}$ and $\theta$ on $\alpha$ for $(n,m,u,v)=(1,1,3,-1)$. (d) Dependence of $\lambda$ and $\delta$ on $\alpha$ for $(1,1,3,-1)$. Dashed lines indicate $a_{rh}$, $\theta$, $\lambda$, and $\delta$ for $\alpha =54.8^{\circ }$ and $\alpha =57.3^{\circ }$. (e) Ball model showing commensurate sites for three sets of indices, $(n,m,u,v)$. Lines show the direction of moiré stripes with respect to the underlying HOPG. The surface unit cells are indicated using dashed lines. Green and cyan shadings in (a)–(d) indicate the maximal ranges of $a_{rh}$ and $\alpha$ that could be consistent with experiment.

Figure 7

CMC model. (a) Ball model for $(n,m,u,v)=(1,1,3,-1)$ ($\alpha =57.3^{\circ }$, $a_{rh}=0.466$ nm, and $\theta =+2.9^{\circ }$). Green atoms in line A are in perfect commensuration with the centers of HOPG hollow sites. Atoms in lines B and C are nearly commensurate with hollow sites of HOPG; for example, green atoms in line B are shifted with respect to underlying HOPG hollow sites by $<1\%$. The red rectangle shows the moiré pattern unit cell defined by two vectors, $(1,1)$ and $(14,-13)$. (b) The region indicated using a black rectangle in (a), shown in more detail. (c) Ball model for $(n,m,u,v)=(1,-1,-1,3)$ with $\theta =-2.9^{\circ }$. Acute and obtuse $\delta$ angles are highlighted in (b) and (c).

Figure 8

Simple superposition model: ball models for Bi unit cell $0.44\times 0.48{\text{nm}}^2$ (star symbols in Fig. 9) and three different misorientation angles $\theta$, (a) $-1$, (b) $-3$, and (c) $+3^{\circ }$. In all images parts of atoms that overlap with the atoms in the graphite substrate are colored white (other atoms are black). $\delta$, $\lambda$, and the main crystallographic directions for the substrate and overlayer are indicated.

Figure 9

Simple superposition model: simulations of Bi slab rotation above graphene substrate showing the dependence of $\delta$ (bottom panel) and $\lambda$ (top panel) on $\theta$ and unit cell selection. Gray symbols are obtained with the symmetry transformation $180-\delta \to \delta$ (see text). Crosshatched regions highlight the experimentally relevant range of values. The $0.45\times 0.48{\text{nm}}^2$ unit cell (green triangles and solid line) is similar to the bulk ($0.454\times 0.475{\text{nm}}^2$). Data from the CMC model ($\alpha =54.8^{\circ }$; see Fig. 6) are shown (diamonds) to allow comparison of the results of the two models.

Figure 10

(a) Model showing moiré pattern formation as a result of distortion at the film-substrate interface (red shading). (b) Ball model of part of the unit cell showing the two atoms [1 (red) and 2 (blue)] that are shifted (along arrows) by 5% during DFT calculations. (c) DFT-calculated band diagrams for 2-ML-thick Bi films for a relaxed unit cell equal to $0.454\times 0.475{\text{nm}}^2$ (black) and distorted unit cells obtained by shifting atoms 1 (red) and 2 (blue) as indicated in (b). (d) Corresponding DOS.