Superlattice Dirac points and space-dependent Fermi velocity in a corrugated graphene monolayer

Recent studies show that periodic potentials can generate superlattice Dirac points at energies $\pm \hslash {\nu }_{\mathrm{F}}\left|\mathbf{G}\right|/2$ in graphene (where $\hslash$ is the reduced Planck constant, ${\nu }_{\mathrm{F}}$ is the Fermi velocity of graphene, and G is the reciprocal superlattice vector). Here, we perform scanning tunneling microscopy and spectroscopy studies of a corrugated graphene monolayer on Rh foil. We show that the quasiperiodic ripples of nanometer wavelength in the corrugated graphene give rise to weak one-dimensional electronic potentials and thereby lead to the emergence of the superlattice Dirac points. The position of the superlattice Dirac point is space dependent and shows a wide distribution of values. We demonstrate that the space-dependent superlattice Dirac points are closely related to the space-dependent Fermi velocity, which may arise from the effect of the local strain and the strong electron-electron interaction in the corrugated graphene.

Figure 1

(a) An STM image of a flat graphene monolayer on Rh foil (V${}_{\mathrm{sample}}$ $=$ 550 mV and I $=$ 6.26 pA). The bright dots around the bottom left corner arise from charge accumulation at the boundary of graphene. (b) Atomic-resolution image of graphene in the blue frame of panel (a) (V${}_{\mathrm{sample}}$ $=$ 600 mV and I $=$ 14.8 pA). The atomic structure of graphene is overlaid onto the STM image. (c) A typical tunneling spectrum, i.e., dI/dV-V curve, recorded on the graphene monolayer. The Dirac point, as marked by the black arrow, is located at $E_{\mathrm{D}}$ ∼ 38 mV, indicating a slight charge transfer between the graphene and substrate. Around the Dirac point, the density of states increases linearly with the energy, which is identical to that in pristine graphene. The dashed and dotted lines are a guide to the eyes.

Figure 2

(a) A STM topography of corrugated graphene with quasiperiodic ripples of nanometer wavelength on Rh foil (V${}_{\mathrm{sample}}$ $=$ 440 mV and I $=$ 10.31 pA). The inset is a Fourier transform of the main panel showing the reciprocal-lattice of the quasiperiodic ripples. The scale bar of the inset is 5 Gm${}^{-1}$. (b) Zoom-in topography of the blue frame in (a) shows a 1D superlattice (V${}_{\mathrm{sample}}$ $=$ 393 mV and I $=$ 10.31 pA). (c) Atomic-resolution image of the red frame in panel (b) shows a honeycomb lattice (V${}_{\mathrm{sample}}$ $=$ 393 mV and I $=$ 10.31 pA). (d) Tunneling spectra, i.e., dI/dV-V curves, recorded at different positions, as marked by the stars with different colors in panel (a). The spectra were vertically offset for clarity. The positions of the superlattice Dirac points, which contribute to the dips of the local density of states, are indicated by the arrows.

Figure 3

(a) An STM topography of a corrugated graphene with quasiperiodic ripples. The dashed lines represent the boundaries of the quasiperiodic ripples. The index $n$ is shown above the ripples. The inset is a Fourier transform of the main panel showing the reciprocal-lattice of the quasiperiodic ripples. (b) A line profile along the black curve in panel (a). (c) Three tunneling spectra, i.e., dI/dV-V curves, recorded at different positions, as marked by the stars with different colors in panel (a). The spectra were vertically offset for clarity. The positions of the superlattice Dirac points, which contribute to the dips of the local density of states, are indicated by the arrows.

Figure 4

(a) The schematic structural model of 1D graphene superlattice formed by a Kronig–Penney type of potential periodic along the $x$ direction with spatial period $L$ and barrier width $W$. (b) Energy dispersions of charge carriers at the minizone boundaries of a 1D graphene superlattice with $U_{1\mathrm{D}}$ $=$ 0.5 eV, $L$ $=$ 10 nm, and $W$ $=$ 5 nm. The gap closes at the minizone boundaries ($k_{\mathrm{x}}=\pm \pi /L$ and $k_{\mathrm{y}}$ $=$ 0) where the superlattice Dirac points are generated. In panel (c), we have a cut of the electronic band structures at a fixed $k_{\mathrm{y}}$. The straight lines correspond to $k_{\mathrm{y}}$ $=$ 0, and the green and blue curves correspond to $k_{\mathrm{y}}$ $=$ 0.012 Å${}^{-1}$. The energy gap at the minizone boundaries is zero for $k_{\mathrm{y}}$ $=$ 0. (d) The energy of the superlattice Dirac points away from the Dirac point as a function of the potential period $L$. The two solid curves are the theoretical dependence of $E_{\mathrm{SD}}\left(L\right)$ with η $=$ 0.95 (blue) and 0.5 (red), respectively (here η $=$ $W$/$L$). The solid circles are the positions of the superlattice Dirac points obtained from the tunneling spectra of Fig. 2d.

Figure 5

(a) Two typical dI/dV-V curves recorded at different positions of the corrugated graphene (the two curves are reproduced from that of Fig. 2 with the same colors) along with the STS spectrum of the flat graphene monolayer. Only the spectra in positive bias are shown for clarity. The arrows point to the positions of the superlattice Dirac points observed in the corrugated graphene. (b) The STS spectrum of the flat graphene monolayer. (c)–(i) Seven dI/dV-V curves of the corrugated graphene from top to bottom in Fig. 2d. Only the spectra in positive bias are shown for clarity. Around the Dirac point, the slope of the spectrum is almost linear in energy. The dashed line is a linear fit to obtain the slope of the spectrum.

Figure 6

(a) The value of $S$, deduced from the spectra at different positions of different corrugated graphene samples, as a function of $E_{SD}^{\exp }/E_{SD}^{\mathrm{th}}$. Here, $S$ is the normalized slope of an STS spectrum, and $S$ $=$ 1 corresponds to the slope of the STS spectrum of pristine graphene, $E_{SD}^{\exp }$ is the corresponding position of the superlattice Dirac points, $E_{SD}^{\mathrm{th}}$ is the theoretical position of the superlattice Dirac points calculated with the width of the ripples $L$ determined experimentally and $W$/$L$ $=$ 0.5. It reveals that the value of $E_{SD}^{\mathrm{th}}$ is closely related to the slope of the STS spectrum. (b) The value of (1/$S$)${}^{0.5}$, deduced from the spectra at different positions of different corrugated graphene samples, as a function of $E_{SD}^{\exp }/E_{SD}^{\mathrm{th}}$. The right $y$ axis shows $v_F^{\mathrm{corr}}/v_F^{\mathrm{flat}}$. The red dashed line is a guide to the eyes.