Electromagnetic absorption and Kerr effect in quantum Hall ferromagnetic states of bilayer graphene

In a quantizing magnetic field, the chiral two-dimensional electron gas in Landau level $N=0$ of bilayer graphene goes through a series of phase transitions at integer filling factors $\nu \in \left[-3,3\right]$ when the strength of an electric field applied perpendicularly to the layers is increased. At filling factor $\nu =3,$ the electron gas can be described by a simple two-level system where layer and spin degrees of freedom are frozen. The gas then behaves as an orbital quantum Hall ferromagnet. A Coulomb-induced Dzyaloshinskii-Moriya term in the orbital pseudospin Hamiltonian is responsible for a series of transitions first to a Wigner crystal state and then to a spiral state as the electric field is increased. Both states have a nontrivial orbital pseudospin texture. In this work, we study how the phase diagram at $\nu =3$ is modified by an electric field applied in the plane of the layers and then derive several experimental signatures of the uniform and nonuniform states in the phase diagram. In addition to the transport gap, we study the electromagnetic absorption and the Kerr rotation due to the excitations of the orbital pseudospin-wave modes in the broken-symmetry states.

Figure 1

Phase diagram of the 2DEG in bilayer graphene at filling factor $\tilde{\nu }=1$. The biases ${\mathrm{\Delta }}_i$ indicate the onset of each phase. (The bias axis is not to scale.)

Figure 2

Electronic density $n\left(\mathbf{r}\right)$ and dipole pattern (arrows) $\mathbf{d}\left(\mathbf{r}\right)$ for the Wigner crystal phase at ${\mathrm{\Delta }}_B=59$ meV.

Figure 3

Electronic density $n\left(\mathbf{r}\right)$. The arrows represent the dipole $\mathbf{d}\left(\mathbf{r}\right)$ and pseudospin fields $〈p_z\left(\mathbf{r}\right)〉$ of the spiral phase for ${\mathrm{\Delta }}_B=112$ meV.

Figure 4

Dispersion relations for the (a) incoherent phase $I$ at ${\mathrm{\Delta }}_B=29$ meV, (b) coherent phase $O$ at ${\mathrm{\Delta }}_B=54$ meV, (c) crystal phase $C$ at ${\mathrm{\Delta }}_B=66$ meV, and (d) spiral phase $S$ at ${\mathrm{\Delta }}_B=77$ meV.

Figure 5

Hartree-Fock electron-hole gap ${\mathrm{\Delta }}_{eh}$ as a function of bias for two values of the parallel electric field ${\mathbf{E}}_{\parallel }=E_{\parallel }\widehat{\mathbf{x}}$ in mV/nm. The dashed lines indicate the gap in the uniform coherent phase $O$ when it is not the ground state. This phase is replaced by the crystal and spiral phases with smaller gap in most of the phase diagram.

Figure 6

Phase diagram of the 2DEG as a function of the bias ${\mathrm{\Delta }}_B$ and the electric field ${\mathbf{E}}_{\parallel }$.

Figure 7

In phase $I,$ a parallel electric field ${\mathbf{E}}_{\parallel }$ tilts the pseudospins towards the x-y plane by an angle $\theta \left({\mathrm{\Delta }}_B,E_{\parallel }\right).$ The field $E_{\parallel }$ in the legend is in mV/nm.

Figure 8

Optical gaps ${\omega }_I\left(0\right)$ and ${\omega }_O\left(0\right)$ as a function of bias for different values of the parallel electric field (in mV/nm)$.$ The flat region of each curve corresponds to the coherent phase $O$.

Figure 9

Dispersion relation of the collective modes in the direction of the spiral at bias ${\mathrm{\Delta }}_B=130$ meV for electric field strengths $E_{\parallel }=0$ and $E_{\parallel }=0.15$ mV/nm.

Figure 10

Absorptions $P_{\pm }\left(\omega \right)$ in the crystal phases $C$ at ${\mathrm{\Delta }}_B=66$ meV [(a) and (b)] and $C^{*}$ at the conjugate bias ${\mathrm{\Delta }}_B=194$ meV [(c) and (d)] with and without a parallel electric field $E_{\parallel }=0.04$ mV/nm.

Figure 11

Absorptions $P_{\pm }\left(\omega \right)$ in the spiral phases at conjugate biases ${\mathrm{\Delta }}_B=77$ meV [(a) and (b)] and ${\mathrm{\Delta }}_B=183$ meV [(c) and (d)] with and without a parallel electric field $E_{\parallel }=0.04$ mV/nm. (e) Absorption at bias ${\mathrm{\Delta }}_M=130$ meV and for $E_{\parallel }=0.$

Figure 12

Kerr angle for conjugate biases ${\mathrm{\Delta }}_B=0$ and ${\mathrm{\Delta }}_B=260$ meV in the $I$ and $I^{*}$ phases.

Figure 13

Kerr angle for conjugate biases (a) ${\mathrm{\Delta }}_B=66$ meV and (b) ${\mathrm{\Delta }}_B=194$ meV in the crystal phase and conjugate biases (c) 77 meV and (d) 183 meV in the spiral phase with a $x$ polarization and (e) with a $y$ polarization of the incident wave.