Strain-induced one-dimensional Landau level quantization in corrugated graphene

Theoretical research has predicted that a ripple of graphene generates an effective gauge field on its low-energy electronic structure and could lead to Landau quantization. Here, we demonstrate using a combination of an experimental method (scanning tunneling microscopy) and a theoretical approach (tight-binding approximation) that Landau levels will form when the effective pseudomagnetic flux per ripple $\Phi \sim (h^2/la){\Phi }_0$ is larger than the flux quantum ${\Phi }_0$ (here, $h$ is the height, $l$ is the width of the ripple, and $a$ is the nearest C-C bond length). The strain-induced gauge field in the ripple only results in one-dimensional (1D) Landau-level quantization along the ripple. Such 1D Landau quantization does not exist in two-dimensional systems in an external magnetic field. Its existence offers a unique opportunity to realize interesting electronic properties in strained graphene.

Figure 1

(a) An STM image of corrugated graphene with 1D ripples showing a wide distribution of periods on Rh foil ($V_{\mathrm{sample}}$ $=$ 1.20 V and $I$ $=$ 0.21 nA). The inset is a Fourier transform of the main panel showing the reciprocal lattice of the ripples. The scale bar is 1 nm${}^{-1}$. (b) A zoom-in topography in the red frame of (a) shows the observed triangular pattern on the ripple ($V_{\mathrm{sample}}=-330$ mV and $I$ $=$ 0.14 nA). The atomic structure of graphene is overlaid onto the STM image. The inset is a Fourier transform of (b). The scale bar is 20 nm${}^{-1}$. (c) Tunneling spectra recorded at different positions, as marked by the dots along the arrow from a to g in (a). The spectra were vertically offset for clarity. (d) The dI/dV-$V$ curves, recorded at different positions, as marked by the dots along the arrow from h to m in (a). A pronounced peak close to the Fermi energy is observed in the tunneling spectra of the ripple.

Figure 2

Fourteen tunnelling curves from top to down are recorded at positions a (curves 1 and 2), b (curves 3 and 4), c (curves 5 and 6), d (curves 7 and 8), e (curves 9 and 10), f (curves 11 and 12), and g (curves 13 and 14). The red dashed lines flanking the Dirac points are used to show the slop of these tunnelling spectra.

Figure 3

(a) Perspective view of two graphene ripples, which can be approximated as a sinusoid. The period and amplitude of the ripples are $l$ and $h$, respectively. (b) The lowest energy band of a graphene ripple with ($h$, $l$) $=$ (0, 21), (0.3, 21), (0.8, 21), and (1, 21), respectively (both $h$ and $l$ are in unit of nm). In the calculation, we only take into account the nearest-neighbor hopping $t_0$ and set the next nearest-neighbor hopping $t^{\prime }$ $=$ 0. (c) The corresponding local DOS of the ripples in (b). (d) The lowest energy band of a graphene ripple with ($h$, $l$) $=$ (1, 21) and ($t_0$, $t^{\prime }$) $=$ (3 eV, 0 eV), (3 eV, 0.06 eV), (3 eV, 0.33 eV), and (3 eV, 0.6 eV), respectively. For clarity, the energy of all the band structure is shifted 3$t^{\prime }$ to make the flat bands stay at zero energy. For the case $t^{\prime }$ ≠ 0, the electronic band structure of the ripple is not symmetry about the zero energy. (e) The corresponding DOS of the ripples in (d). For clarity, the energy of all the DOS is shifted 3$t^{\prime }$ to make the peak of the DOS stays at zero energy. When $t^{\prime }$ ≠ 0, the slope of the DOS flanking the zero energy is not symmetry and the asymmetry increases with increasing the $t^{\prime }$.

Figure 4

(a) The ratio of the slope |$S_R$/S${}_L$| in the calculated density of states as a function of $t^{\prime }/t_0$. (b) The ratio of the slope |$S_R$/S${}_L$| of the tunnelling spectra obtained in our experiment. The dotted line shows the average ratio |$S_R$/S${}_L$| ∼ 1.25.

Figure 5

(a) Energy dispersions of the graphene ripples. (b) Energy dispersions of the 1D graphene superlattice. Both band structures show anisotropy of the Fermi velocity.

Figure 6

(a) A typical dI/dV-$V$ curve (dark curve) of the graphene ripple [this curve is reproduced from that of Fig. 1d]. The red asterisks denote the positions of Landau levels. (b) The energy of pseudo-Landau level peaks $E_n$-$E_D$ deduced from the spectrum in (a) as a function of sgn($n$)|$n$|${}^{1/2}$. The red dashed line is the linear fit. The inset shows a schematic energy band structure of the ripple, which shows linear energy dispersion in the $k_x$ direction and the Landau level quantization in the $k_y$ direction.

Figure 7

This figure summarizes the parameters, $l$ and $h$, of the ripples studied in our experiment. The filled circle represents the ripple showing Landau levels and the open circles are denoted the ripples that not show any evidence of the Landau level. The red dashed curve is plotted according to $h^2/la=1$.