Surface states and quasiparticle interference in Bernal and rhombohedral graphite with and without trigonal warping

We use an exact analytical technique [Phys. Rev. B 101, 115405 (2020), Phys. Rev. B 102, 165117 (2020)] to recover the surface Green's functions for Bernal (ABA) and rhombohedral (ABC) graphite. For rhombohedral graphite we recover the predicted surface flat bands. For Bernal graphite we find that the surface state spectral function is similar to the bilayer one, but the trigonal warping effects are enhanced, and the surface quasiparticles have a much shorter lifetime. We subsequently use the $T$-matrix formalism to study the quasiparticle interference patterns generated on the surface of semi-infinite ABA and ABC graphite in the presence of impurity scattering. We compare our predictions to experimental STM data of impurity-localized states on the surface of Bernal graphite which appear to be in a good agreement with our calculations.

Figure 1

Lattice structure for A: Single-layer graphene, B: Bilayer graphene with AB stacking, C: Bernal-stacked graphite, D: Rhombohedral-stacked graphite. Modified from Ref. [10].

Figure 2

Spectrum for Bernal-stacked graphite around the $K$ point for 100 layers, with ${\gamma }_3=0$.

Figure 3

Spectrum for ABC graphite around the $K$ point for 10, 50, and 100 layers (from left to right), and ${\gamma }_3=0$.

Figure 4

A planelike impurity (in black) separating a 3D infinite system into two semi-infinite parts, each with a surface (in red) parallel to the impurity plane. The impurity and the red surfaces are separated by a unit-cell spacing, which in this case is equal to $2d_0$ for Bernal-stacked graphite and to $3d_0$ for rhombohedral-stacked graphite.

Figure 5

Surface spectral function of Bernal-stacked graphite around the $K$ point, at $z=2d_0$. We have taken ${\gamma }_3=0$ and $\delta =0.005$.

Figure 6

Surface spectral functions of bilayer graphene (left column) and Bernal-stacked graphite (right column), around the $K$ point calculated at $E=0.05$, without (top row, ${\gamma }_3=0$) and with trigonal warping (bottom row, ${\gamma }_3=-0.3$). We have taken $\delta =0.005$.

Figure 7

Surface spectral function of rhombohedral-stacked graphite around the $K$ point, at $z=3d_0$. We have taken ${\gamma }_3=0$ and $\delta =0.005$.

Figure 8

Spectral function of trilayer ABC graphene (left column) and surface spectral function for ABC graphite (right column), without trigonal warping ${\gamma }_3=0$ (top row) and with trigonal warping ${\gamma }_3=-0.3$ (bottom two rows) around the $K$ point at $E=0$ (top two rows) and $E=0.05$ (bottom row). We have taken $\delta =0.005$.

Figure 9

Quasiparticle interference patterns for bilayer graphene (left panel) and Bernal-stacked graphite (right panel) around the $K$ point calculated at $E=0.05$. We consider a trigonal warping value ${\gamma }_3=-0.3$, and we take $V=-10,{\gamma }_0=-3.3$.

Figure 10

Quasiparticle interference features for ABC graphene (left) and rhombohedral-stacked graphite (right) around the $K$ point at $E=0$ (top row), and $E=0.05$ (bottom row). We take $\delta =0.005$, ${\gamma }_3=-0.3$, and $V=-33$.

Figure 11

Impurity-induced corrections to the local density of states in bilayer graphene (left column row) and Bernal-stacked graphite (right column) calculated at $E=0.05$. We set $V=-33$ and ${\gamma }_3=-0.3$. The four rows correspond to impurities located on sites A1/B1/A2/B2. The LDOS corrections are given in units of the value of the unperturbed background. The results for bilayer graphene are in a good agreement with the analytical calculations presented in Ref. [44].

Figure 12

Impurity-induced corrections to the local density of states in ABC graphite at $E=0$, ${\gamma }_3=-0.3$, $V=-33$, and $\delta =0.005$. The panels correspond to impurities located on sites A1/B1/A2/B2/A3/B3. The LDOS corrections are given in units of the value of the unperturbed background.

Figure 13

Experimental measurements of the real-space and Fourier-space quasiparticle interference patterns for impurities in graphite and many-layer graphene. (a) and (b) $dI/d{V}_S$ maps obtained on, respectively, graphite and ten-layer-thick graphene film supported by hexagonal boron nitride. (c) and (d) Zoom-ins around the origin of the respective fast Fourier transforms. The $dI/dV$ maps were measured at a tip-sample bias $V_S=+5\mathrm{mV}$, while applying a $V_{\mathrm{AC}}=3.5\mathrm{mV}$ excitation with a lock-in amplifier, and scanning at a constant current $I=0.5\mathrm{nA}$ and $I=0.25\mathrm{nA}$ for (a) and (b), respectively.

Figure 14

Impurity-induced corrections to the local density of states in bilayer graphene at four different energies, $E=0.25$ (top left), 0.05 (top right), 0.02 (bottom left), and 0 (bottom right). We take ${\gamma }_3=-0.3$, $\delta =0.005$, and $V=-33$. The LDOS corrections are given in units of the value of the unperturbed background.

Figure 15

Impurity-induced corrections to the local density of states in bilayer graphene (top right) and in a three-layer (top left), four-layer (bottom left), and eight-layer (bottom right) systems at $E=0.05$. We take ${\gamma }_3=-0.3$, $\delta =0.005$, and $V=-33$. The LDOS corrections are given in units of the value of the unperturbed background.

Figure 16

Impurity-induced corrections to the local density of states in an eight-layer system at $E=0$. We take ${\gamma }_3=-0.3$, $\delta =0.005$, and $V=-33$. The LDOS corrections are given in units of the value of the unperturbed background.

Figure 17

Left: Equal-energy contours of bilayer graphene electronic bands plotted for lower and higher energies (red and blue contours, respectively). Right: Bands of bilayer graphene. We take $a_0=2.46$ Å, ${\gamma }_0=3.3\mathrm{eV}$, ${\gamma }_1=0.42\mathrm{eV}$, ${\gamma }_3=-0.3\mathrm{eV}$. The lower-energy contours are calculated at $E=0.0007$, 0.0014, and $0.002\mathrm{eV}$, while the higher-energy contours are at $E=0.01$, 0.02, and $0.03\mathrm{eV}$.