Detecting giant electron-hole asymmetry in a graphene monolayer generated by strain and charged-defect scattering via Landau level spectroscopy

The electron-hole symmetry in graphene monolayer, which is analogous to the inherent symmetric structure between electrons and positrons of the Universe, plays a crucial role in the chirality and chiral tunneling of massless Dirac fermions. Here we demonstrate that both strain and charged-defect scattering could break this symmetry dramatically in a graphene monolayer. In our experiment, the Fermi velocities of electrons $v_F^e$ and holes $v_F^h$ are measured directly through Landau level spectroscopy. In strained graphene with lattice deformation and curvature, the $v_F^e$ and $v_F^h$ are measured as $(1.21\pm 0.03)\times {10}^6\mathrm{m}/\mathrm{s}$ and $(1.02\pm 0.03)\times {10}^6\mathrm{m}/\mathrm{s}$, respectively. This giant asymmetry originates from enhanced next-nearest-neighbor hopping in the strained region. Around positively charged defect, we observe opposite electron-hole asymmetry, and the $v_F^e$ and $v_F^h$ are measured to be $(0.86\pm 0.02)\times {10}^6\mathrm{m}/\mathrm{s}$ and $(1.14\pm 0.03)\times {10}^6\mathrm{m}/\mathrm{s}$, respectively. Such a large asymmetry is attributed to the fact that the massless Dirac fermions in a graphene monolayer are scattered more strongly when they are attracted to the charged defect than when they are repelled from it.

Figure 1

(a) An atomic-resolution STM image of a graphene sheet on Rh foils ($V_{\mathrm{sample}}=0.57\mathrm{V}$ and $I=69\mathrm{pA}$). The inset is the fast Fourier transform showing the reciprocal lattice of graphene. The scale bar is $15\mathrm{G}{\mathrm{m}}^{-1}$. (b) The height profile along the dashed blue line in (a) indicating ultrafine rippling of the graphene sheet. (c) STS spectra taken under different magnetic fields at the green solid circle marked in panel (a). For clarity, the curves are offset in the $Y$ axis and LL indices of massless Dirac fermions are marked. (d) LL peak energies of the spectra plotted against $\mathrm{sgn}(n){\left|nB\right|}^{1/2}$. The red solid line is a linear fit of the data with Eq. (1). Inset: the schematic LL structure of the graphene monolayer.

Figure 2

(a) A representative STM image showing quasiperiodic graphene ripples. (b) STM image of a graphene ripple in the white frame in (a) ($V_{\mathrm{sample}}=0.57\mathrm{V}$ and $I=69\mathrm{pA}$). The solid blue line is the height profile across the rippled graphene region. The inset is the fast Fourier transform showing the reciprocal lattice of the graphene lattice in the red frame. The scale bar is $20\mathrm{G}{\mathrm{m}}^{-1}$. (c) STS spectra taken at the light blue solid circle marked in (b) under different magnetic fields. (d) LL peak energies plotted against $\mathrm{sgn}(n){\left|nB\right|}^{1/2}$. The red and blue solid lines are linear fits of the data with Eq. (1) for electrons and holes, respectively. Inset: the schematic LLs structure of the strained graphene sheet. The $n>0$ LLs in dotted curves are plotted to show the LL structure with electron-hole symmetry. (e) Left: lattice structure of graphene. ${\vec{a}}_1$ and ${\vec{a}}_2$ are the lattice unit vectors. Right: corresponding Brillouin zone. ${\vec{b}}_1$ and ${\vec{b}}_2$ are the reciprocal-lattice unit vectors. (f) Electronic dispersion for a graphene honeycomb lattice with different $t^{\prime }(t=3\mathrm{eV})$.

Figure 3

(a) A representative STM image of a graphene monolayer with three atomic defects ($V_{\mathrm{sample}}=0.98\mathrm{V}$ and $I=290\mathrm{pA}$), as marked by the three dotted circles. The graphene sheet also exhibits nonuniform nanoscale ripples. The wavelength and amplitude of these nanoscale ripples are shown in Fig. S4. The inset shows an enlarged image in the white frame showing the honeycomb lattice and the scale bar is $0.3\mathrm{nm}$. (b) The zoom-in STM image of the defect in the red dotted circle in panel (a). (c) Normalized STS spectra, $(dI/dV)/(I/V)-V$, measured on graphene at different distances from the center of the defect. (d) STS spectra taken at the red solid dot marked in (a) under different magnetic fields. The blue arrows indicate the resonance peaks of the defect under different magnetic fields. $E_D$ is the Dirac point and the $n=0\mathrm{LL}$ is marked.

Figure 4

(a) STS spectra taken under different magnetic fields at the blue solid circle marked in Fig. 3. The blue arrows indicate the resonance peak of the defect under different magnetic fields. (b) shows the values of $E_p–E_0$ obtained at different positions as a function of the magnetic fields. The blue dashed line is a guide to the eyes. (c) LL peak energies in panel (a) plotted against $\mathrm{sgn}\left(n\right){\left|nB\right|}^{1/2}$. The red and blue solid lines are the linear fits of the data with Eq. (1). Inset: the schematic LL structure of the graphene sheet around the charged defect. The $n<0$ LLs in dotted curves are plotted to show the LL structure with electron-hole symmetry. (d) This figure summarizes the Fermi velocities of electrons and holes obtained at different positions around the charged defects. The inset shows the STM image with 13 different positions marked by the light blue numbers. (e) Transport cross section $C$ as a function of the impurity charge ${\mathbf{\alpha }}_{\mathbf{\varepsilon }}$. Impurity charge ${\mathbf{\alpha }}_{\varepsilon }=\alpha \mathrm{sgn}\left(\varepsilon \right)$, where ${\mathbf{\alpha }}_{\varepsilon }>0({\mathbf{\alpha }}_{\varepsilon }<0)$ indicates the attractive (repulsive) interaction between the impurity and the carriers.