Characterization of graphene through anisotropy of constant-energy maps in angle-resolved photoemission

We theoretically show how constant-energy maps of the angle-resolved photoemission intensity can be used to test wave function symmetry in graphene. For monolayer graphene, we demonstrate that the observed anisotropy of angle-resolved photoelectron spectroscopy spectra is a manifestation of what has been recently branded as an electronic chirality. For bilayer graphene, we show that the anisotropy of the constant-energy maps may be used to extract information about the magnitude and sign of interlayer coupling parameters and about symmetry breaking inflicted on a bilayer by the underlying substrate.

Figure 1

(a) Schematic of the monolayer lattice containing two sites in the unit cell: $A$ (white circles) and $B$ (black circles). (b) Schematic of the hexagonal and rhombic Brillouin zone indicating two inequivalent valleys $K_{\pm }$ showing the wave vector $\mathbf{q}=\left(q\text{cos}\phi ,q\text{sin}\phi \right)$ measured from the center of valley $K_{+}$. (c) Schematic of the low energy bands $ϵ\approx \pm \hslash vq$ near the $K_{+}$ point obtained by taking into account intralayer hopping with velocity $v$. The shading indicates the region of occupied states up to the Fermi energy ${ϵ}_F$ and the dashed line indicates a typical energy of states contributing to photoemission, whereby incoming photons of energy $\hslash \omega$ produce photoelectrons of kinetic energy $E_{\mathbf{p}}$. (d) The intensity of photoemission from states at a constant energy of 1.4 eV below the charge-neutrality point (Ref. 38) in monolayer graphene, which is plotted as a function of photoelectron wave vector ${\mathbf{p}}_{\parallel }=\left(p_x,p_y\right)$ parallel to the surface of graphene for ${\mathbf{p}}_{\parallel }$ covering several Brillouin zones (top) and plotted as a function of photoelectron wave vector $\mathbf{q}=\left(q_x,q_y\right)$ in the vicinity of valley $K_{+}$ (bottom) [note that the origin and scale of ${\mathbf{p}}_{\parallel }$ and $\mathbf{q}$ are different]. Here, we use the parameter values ${\gamma }_0=3.0\text{eV}$, $s_0=0.129$, $\Delta =0$, and energy width $\Gamma =0.24\text{eV}$.

Figure 2

(a) Schematics of the low-energy bands $ϵ=\pm \hslash vq$ near the $K_{+}$ point in the absence of intralayer asymmetry $\Delta$. (b) The intensity of photoemission from states at a fixed energy close to the charge-neutrality point in monolayer graphene, which is plotted as a function of photoelectron wave vector ${\mathbf{q}}_{\parallel }=\left(q_x,q_y\right)$ parallel to the surface of graphene in the vicinity of valley ${\mathbf{p}}_{\parallel }=\left(4\pi /3a,0\right)$. Each plot corresponds to a different energy with respect to the charge-neutrality point either above (left-hand side) or below (right-hand side). The units of relative intensity (from 0 to 1) are chosen to illustrate the anisotropy, with different integral intensities in different rows. The parameter values are ${\gamma }_0=3.0\text{eV}$, $s_0=0.129$, and $\Delta =0$, and the energy width $\Gamma$ varies as the energy is divided by 6. (c) The development of the intensity pattern for emission at a fixed energy of 0.1 eV below the charge-neutrality point in the vicinity of valley ${\mathbf{p}}_{\parallel }=\left(4\pi /3a,0\right)$ in monolayer graphene as intralayer asymmetry increases in magnitude as $\Delta /2=0.05$, 0.075, and 0.1 eV. The parameter values are ${\gamma }_0=3.0\text{eV}$ and $s_0=0.129$, and the Lorentzian energy broadening is $\Gamma =0.0167$.

Figure 3

(a) Schematic of the bilayer lattice containing four sites in the unit cell: $A1$ (white circles) and $B1$ (black circles) at the bottom layer and $A2$ (white circles) and $B2$ (black circles) at the top layer. (b) Schematic of the hexagonal and rhombic Brillouin zone indicating two inequivalent valleys $K_{\pm }$ showing the wave vector $\mathbf{q}=\left(q\text{cos}\phi ,q\text{sin}\phi \right)$ measured from the center of valley $K_{+}$. (c) Schematic of the low energy bands in the absence of lattice asymmetry. The energy band index $\alpha =\pm 1$ [Eq. (11)] is explicitly shown for the case ${\gamma }_1<0$. (d) The intensity of photoemission from states at a constant energy of 1.45 eV below the charge-neutrality point in bilayer graphene, which is plotted as a function of photoelectron wave vector ${\mathbf{p}}_{\parallel }=\left(p_x,p_y\right)$ parallel to the surface of graphene for ${\mathbf{p}}_{\parallel }$ covering the whole Brillouin zone (left) and plotted as a function of photoelectron wave vector $\mathbf{q}=\left(q_x,q_y\right)$ in the vicinity of valley $K_{+}$ (right) (note that the origin and scale of ${\mathbf{p}}_{\parallel }$ and $\mathbf{q}$ are different). Here, we use the parameter values ${\gamma }_0=3.0\text{eV}$, ${\gamma }_1=-0.35\text{eV}$, ${\gamma }_3=-0.15\text{eV}$, ${\gamma }_4=0.0\text{eV}$, $s_0=0.129$, and $\Delta =U=0$, and energy width $\Gamma =0.24\text{eV}$.

Figure 4

The intensity of photoemission from states at a fixed energy close to the charge-neutrality point in bilayer graphene, which is plotted as a function of photoelectron wave vector $\mathbf{q}=\left(q_x,q_y\right)$ parallel to the surface of graphene in the vicinity of valley ${\mathbf{p}}_{\parallel }=\left(4\pi /3a,0\right)$. Each plot corresponds to a different energy with respect to the charge-neutrality point either above (left) or below (right). Units of relative intensity (from 0 to 1) are chosen to illustrate the anisotropy, with different integral intensities in different rows. The parameter values are ${\gamma }_0=3.0\text{eV}$, ${\gamma }_1=-0.35\text{eV}$, ${\gamma }_3=-0.15\text{eV}$, ${\gamma }_4=0.0\text{eV}$, $s_0=0.129$, and $\Delta =U=0$, and the energy width $\Gamma$ varies as the energy is divided by 6.

Figure 5

The intensity of photoemission in bilayer graphene for fixed energy very close to the charge-neutrality point in the vicinity of valley ${\mathbf{p}}_{\parallel }=\left(4\pi /3a,0\right)$: for states with an energy of 0.03 eV above the charge-neutrality point (left) and states with an energy of 0.03 eV below the charge-neutrality point (right). We consider different signs of the $A2\text{−}B1$ interlayer coupling strength ${\gamma }_1$, with ${\gamma }_1=-0.35\text{eV}$ (top) and ${\gamma }_1=+0.35\text{eV}$ (bottom). The other parameter values are ${\gamma }_0=3.0\text{eV}$, ${\gamma }_3=-0.15\text{eV}$, ${\gamma }_4=0.0\text{eV}$, $\Gamma =0.005\text{eV}$, and $s_0=0.129$ (Ref. 39).

Figure 6

The band structure of bilayer graphene in the vicinity of a valley for no lattice asymmetry (left), substrate-induced asymmetry $\Delta ={\gamma }_1<0$ $\left(U=0\right)$ (center), and interlayer asymmetry $U={\gamma }_1/2$ $\left(\Delta =0\right)$ (right). For clarity, we use large values of asymmetry. The energy band index $\alpha =\pm 1$ [Eq. (11)] is explicitly shown for the case ${\gamma }_1<0$ on the left-hand side.

Figure 7

Left (right) shows the development of the intensity pattern for emission at a fixed energy of 0.1 eV above (below) the midgap energy in bilayer graphene in the vicinity of valley ${\mathbf{p}}_{\parallel }=\left(4\pi /3a,0\right)$ as intralayer asymmetry $\left|\Delta \right|$ increases [the energy with respect to the charge-neutrality point is also indicated]. The plots show $\Delta =0$ (top), negative $\Delta /2=-0.15\text{eV}$ (middle), and positive $\Delta /2=0.15\text{eV}$ (bottom). The same scale of relative intensity (from 0 to 1) is used in all graphs. The parameter values are ${\gamma }_0=3.0\text{eV}$, ${\gamma }_1=-0.35\text{eV}$, ${\gamma }_3=-0.15\text{eV}$, ${\gamma }_4=0.0\text{eV}$, $s_0=0.129$, and $\Gamma =0.0167\text{eV}$.

Figure 8

The development of the intensity pattern in bilayer graphene in the vicinity of valley ${\mathbf{p}}_{\parallel }=\left(4\pi /3a,0\right)$ as interlayer asymmetry $U/2=0$, 0.075, 0.1, 0.125, and 0.15 eV increases for emission from states with an energy of 0.1 eV above the charge-neutrality point on the left-hand side and an energy of 0.1 eV below the charge-neutrality point on the right-hand side. The same scale of relative intensity (from 0 to 1) is used in all graphs. The parameter values are ${\gamma }_0=3.0\text{eV}$, ${\gamma }_1=-0.35\text{eV}$, ${\gamma }_3=-0.15\text{eV}$, ${\gamma }_4=0.0\text{eV}$, $s_0=0.129$, and $\Gamma =0.0167\text{eV}$.

Figure 9

The development of the intensity pattern in bilayer graphene for emission from states with an energy of 0.1 eV below the charge-neutrality point in the vicinity of valley ${\mathbf{p}}_{\parallel }=\left(4\pi /3a,0\right)$ as attenuation [described by the factor $\text{exp}\left(-2z+2i\beta \right)$, where $\left(z,\beta \right)$ are real parameters] of waves from the bottom layer increases. The parameter values are ${\gamma }_0=3.0\text{eV}$, ${\gamma }_1=-0.35\text{eV}$, ${\gamma }_3=-0.15\text{eV}$, ${\gamma }_4=0.0\text{eV}$, $s_0=0.129$, and $\Gamma =0.0167\text{eV}$.