Realizing One-Dimensional Electronic States in Graphene via Coupled Zeroth Pseudo-Landau Levels

Strain-induced pseudomagnetic fields can mimic real magnetic fields to generate a zero-magnetic-field analog of the Landau levels (LLs), i.e., the pseudo-Landau levels (PLLs), in graphene. The distinct nature of the PLLs enables one to realize novel electronic states beyond what is feasible with real LLs. Here, we show that it is possible to realize exotic electronic states through the coupling of zeroth PLLs in strained graphene. In our experiment, nanoscale strained structures embedded with PLLs are generated along a one-dimensional (1D) channel of suspended graphene monolayer. Our results demonstrate that the zeroth PLLs of the strained structures are coupled together, exhibiting a serpentine pattern that snakes back and forth along the 1D suspended graphene monolayer. These results are verified theoretically by large-scale tight-binding calculations of the strained samples. Our result provides a new approach to realizing novel quantum states and to engineering the electronic properties of graphene by using localized PLLs as building blocks.

Figure 1

(a) and (b) Depiction of the top (bottom) layer etched edges. (c) A STM image of top etched sharp edges, $V_s=1.4\mathrm{V}$, $I=0.6\mathrm{nA}$. Inset: height profile across the edge. (d) Close-up atomic and FFT image of the square area in (c), $V_s=0.4\mathrm{V}$, $I=0.5\mathrm{pA}$. Outer spots are the reciprocal lattices of graphene. The inner pattern is induced by intervalley scattering. Scale bar: $3.5{\mathrm{nm}}^{-1}$. (e) A STM image of the bottom etched smooth channel, $V_s=-0.8\mathrm{V}$, $I=0.2\mathrm{pA}$. (f) Close-up atomic and FFT image of the square area in (e). $V_s=0.6\mathrm{V}$, $I=0.1\mathrm{n}\mathrm{A}$. Scale bar: $3.5{\mathrm{nm}}^{-1}$.

Figure 2

(a) A STM image of 1D channels on steps, $V_s=-1.8\mathrm{V}$, $I=0.1\mathrm{nA}$. Scale bar: 20 nm. (b) Two-dimensional projection of the longest channel in (a). The heads of ripples are labeled with black dots and numbers. Note that the ripples labeled as 4 and 8 are difficult to be observed due to the small height. $V_s=0.05\mathrm{V}$, $I=0.015\mathrm{nA}$. Scale bar: 10 nm. Inset: the height profile along the black arrow. The square frames of the insets show the triangular (hexagonal) lattice image on (off) the ripple. $V_s=0.1\mathrm{V}$, $I=0.04\mathrm{nA}$. Scale bar: 0.2 nm. (c) Close-up image of the black frame in (b). Scale bar: 3 nm. (d) Top panel is the $dI/dV$ spectrum taken at the pink dot in (b). The PMF is about 29 T, and the full width at half maximum (FWHM) of the zeroth PLL is 75 meV. Middle panel is the $dI/dV$ spectrum taken at the purple dot in (c). The splitting and FWHM of the zeroth PLLs are 56 and 110 meV, respectively. Bottom panel is the $dI/dV$ spectrum taken at the black dot in (c). The splitting and FWHM of the zeroth PLLs are 62 and 112 meV, respectively. The purple and red solid lines are the fittings of the PLLs. (e) The distribution of PMFs in region I, which is extracted from experimental STS curves.

Figure 3

$dI/dV$ maps measured at the energy of (a) $-0.1\mathrm{eV}$, (b) 0.05 eV and 0.1 eV (around the zeroth PLL), (c) 0.4 eV and 0.6 eV. (d)–(f) Theoretical calculated LDOS maps at the corresponding energies of $-0.1\mathrm{eV}$, 0.05 eV, 0.1 eV, 0.4 eV, and 0.5 eV. Scale bar: 10 nm. All images in the experiment or simulation have the same color scale.

Figure 4

(a) Schematic representation of the calculated graphene monolayer embedded with buckling and periodic ripples that have crests at an angle to the trench. The unit cell is marked with a white dotted frame. Inset: the armchair configuration along the channel. (b) and (e) Contour plots of the calculated average strain $ϵ$ of strained graphene structures 1 and 2, respectively. The pink areas denote the ranges of the strain. (c) and (f) Calculated PMFs of structures 1 and 2. (d) and (g) Calculated LDOS of points 1 and 2 along ripples of structures 1 and 2, respectively. The corresponding positions are marked by black and red dots in (a). LLs indices are marked in figures.