Role of pseudospin in quasiparticle interferences in epitaxial graphene probed by high-resolution scanning tunneling microscopy

Pseudospin, an additional degree of freedom emerging in graphene as a direct consequence of its honeycomb atomic structure, is responsible for many of the exceptional electronic properties found in this material. This paper is devoted to providing a clear understanding of how graphene's pseudospin impacts the quasiparticle interferences of monolayer (ML) and bilayer (BL) graphene measured by low-temperature scanning tunneling microscopy and spectroscopy. We have used this technique to map, with very high energy and space resolution, the spatial modulations of the local density of states of ML and BL graphene epitaxially grown on SiC(0001), in presence of native disorder. We perform a Fourier transform analysis of such modulations including wave vectors up to unit vectors of the reciprocal lattice. Our data demonstrate that the quasiparticle interferences associated to some particular scattering processes are suppressed in ML graphene, but not in BL graphene. Most importantly, interferences with $2q_F$ wave vector associated to intravalley backscattering are not measured in ML graphene, even on the images with highest resolution where the graphene honeycomb pattern is clearly resolved. In order to clarify the role of the pseudospin on the quasiparticle interferences, we use a simple model which nicely captures the main features observed in our data. The model unambiguously shows that graphene's pseudospin is responsible for such suppression of quasiparticle interference features in ML graphene, in particular for those with $2q_F$ wave vector. It also confirms scanning tunneling microscopy as a unique technique to probe the pseudospin in graphene samples in real space with nanometer precision. Finally, we show that such observations are robust with energy and obtain with great accuracy the dispersion of the $\pi$ bands for both ML and BL graphene in the vicinity of the Fermi level, extracting their main tight-binding parameters.

Figure 1

(a) The honeycomb structure of monolayer (ML) graphene. The unit cell includes one carbon atom on each $A$ and $B$ sublattice (two C atoms per unit cell). (b) The first Brillouin zone and the Fermi surface of $n$-doped ML graphene. Inset: The low-energy band structure of ML graphene (Dirac cone), with a linear and isotropic dispersion at $K$ or $K^{\prime }$ points. $E_D$ and $E_F$ are, respectively, the Dirac and Fermi energies. (c) and (d) Schematic of the elastic scattering events in doped ML/BL graphene, divided into intravalley (c) and intervalley (d) processes. The arrows correspond to the wave vectors of the associated LDOS modulations. (e) Schematic of the expected two-dimensional Fourier transform map of the LDOS, including all the processes depicted in (c) and (d), and neglecting the pseudospin (see the main text).

Figure 2

(a) A 100 $\times$ 100 nm${}^2$ constant-current STM image on ML graphene on SiC(0001). Sample bias: $-4$ mV. Number of pixels: 2048 $\times$ 2048. (b) A 5 $\times$ 5 nm${}^2$ numerical zoom of (a), showing a hexagonal atomic pattern of period 0.24 nm characteristics of the graphene's honeycomb structure. A long-range periodic superstructure (period 1.9 nm) is also present, inferred to the interface with the buffer layer (see text). (c) Two-dimensional fast Fourier transform (2D FFT) of (a). Image size: 40 $\times$ 40 nm${}^{-2}$. (d) Central area of (c) showing the absence of intensity ring with radius $2q_F$. (e)–(g) Zoom-in on the three $2q_F$ outer rings in (c) indicated by arrows. Image sizes are 5 $\times$ 5 nm${}^{-2}$ for (d)–(g).

Figure 3

(a) Detail of the 2D FFT of a 150 $\times$ 150 nm${}^2$ constant-current STM image obtained at sample bias $-10$ mV on ML graphene on SiC(0001). Image size 40 $\times$ 40 nm${}^{-2}$. The center $k=0$ of the FFT is labeled (0,0), and first-order spots of the reciprocal lattice are labeled (1,0) and (0,1). The arrows point towards faint rings of radius $2q_F$ centered at these two points. (b) A 5 $\times$ 5 nm${}^2$ numerical zoom of the 150 $\times$ 150 nm${}^2$ image.

Figure 4

(a) Sample-bias dependence of a $2q_E$ ring at $K$ point in the FT-LDOS maps, obtained from a series of 50 $\times$ 50 nm${}^2$ $dI/dV$ images (not shown). Each image of (a) has a size 5 $\times$ 5 nm${}^{-2}$. (b) Dispersion relation $E\left(q_E\right)$ extracted from the radial average of the rings shown in (a). A linear fit is displayed in plain lines, yielding to an estimation of the Fermi velocity $v_F$ and of the Dirac energy $E_D$. (c) A typical $dI/dV\left(V\right)$ spectrum obtained at fixed tip position, with open feedback loop. The arrow points towards a shallow minimum of the conductance curve at sample bias corresponding to the Dirac energy $E_D$ derived in (b). Stabilization parameters: Sample bias, $+$ 350 mV; tunneling current, 0.15 nA.

Figure 5

(a) Schematic Fermi surface of $n$-doped ML graphene, featuring the pseudospin orientations (red arrows) in the two unequivalent valleys at $K$ and $K^{\prime }$ points. (b) Schematic of the Dirac cone at the $K$ point showing the pseudospin orientations (red arrows) for states at energies above and below the Dirac point. (c) Expected FT-LDOS map taking into account the FS topology and the pseudospin (see text).

Figure 6

Illustration of the QIs associated to intravalley scattering off a single atomically sharp impurity. (a) Simulation of the $2q_F$ LDOS oscillation generated on one sublattice only (sublattice $A$), with the impurity located at the center of the image on a site $A$. Image size 100 $\times$ 100 nm${}^2$, 2048 $\times$ 2048 pixels. (b) Numerical zoom of the area limited by the dashed box on (a). (c) Lateral profile along a row of A atoms performed on (b). (d) 2D FFT of (a). (e) same representation as (a), considering both $A$ and $B$ sublattices, with $2q_F$ oscillations shifted by $\pi$ between the two sublattices. No long-range oscillation shows up on this image, in contrast with (a). (f) Numerical zoom of the area limited by the dashed box on (e). (g) Lateral profiles performed on (f) along a row of A atoms and along a row of B atoms. (h) 2D-FFT of (e). See Ref. 72 for the exact quantity mapped on (a) and (e).

Figure 7

(a), (b) Schematic representations of the low-energy band structure and of the pseudospin texture (red arrows) in (a) monolayer graphene and (b) bilayer graphene. (c) Band structure of a standard 2D electron gas. Note that in this latter case, the band is centered on the $\Gamma$ point of the Brillouin zone.

Figure 8

(a) A 5 $\times$ 5 nm${}^2$ numerical zoom of a 50 $\times$ 50 nm${}^2$ constant-current STM image (not shown) recorded on a BL terrace, at sample bias $-25$ mV. The atomic triangular pattern on (a) is the hallmark for bilayer graphene with Bernal stacking. (b) A 50 $\times$ 50 nm${}^2$ $dI/dV$ map of the BL terrace at sample bias $-25$ mV. (c) 40 $\times$ 40 nm${}^{-2}$ 2D FFT image calculated from (b). Both intravalley and intervalley $2q_E$ rings show up, respectively, at the center and at $K$ ($K^{\prime }$) points of the diagram.

Figure 9

(a) Top row: Series of 50 $\times$ 50 nm${}^2$ $dI/dV$ maps of a BL terrace at different sample biases. Middle and bottom rows: corresponding 2D FFT maps, magnified respectively on a $2q_E$ ring at $K$ point and on the central $2q_E$ ring. Images size: 5 $\times$ 5 nm${}^{-2}$. (b) Dispersion relation extracted from the average radius of the rings displayed on (a). The plain curve is a fit of the data for an $n$-doped asymmetric bilayer, with parameters given in the table, close to the parameters derived from ARPES measurements (Ref. 79). (c) A $dI/dV\left(V\right)$ spectrum obtained at fixed tip position, with open feedback loop. A dip of width 0.1 eV centered at $-0.3$ eV reflects the energy band gap induced by the doping asymmetry between the two graphene layers. Stabilization parameters: sample bias, $+$ 350 mV; tunneling current, 0.15 nA.

Figure 10

Summary table of the main results obtained in this study. (a) Schematic pseudospin texture of ML. (b) Schematic FT LDOS map taking into account the Fermi surface topology and the pseudospin of ML, derived from our model. (c) FT LDOS map obtained from STM measurements on ML graphene on SiC(0001) at 5 K. Note the good agreement between (b) and (c). (d) Dispersion relation derived from the STM data on ML. (e)–(h) same as (a)–(d) but for BL graphene. Note the correspondence between (f) and (g): the pseudospin has no significant impact for the BL, contrary to the ML case.