Theory of inter-Landau-level magnetoexcitons in bilayer graphene

If bilayer graphene is placed in a strong perpendicular magnetic field, several quantum Hall plateaux are observed at low enough temperatures. Of these, the ${\sigma }_{xy}=4n{e}^2/h$ sequence ($n\ne 0$) is explained by standard Landau quantization, while the other integer plateaux arise due to interactions. The low-energy excitations in both cases are magnetoexcitons, whose dispersion relation depends on single- and many-body effects in a complicated manner. Analyzing the magnetoexciton modes in bilayer graphene, we find that the mixing of different Landau level transitions not only renormalizes them, but essentially changes their spectra and orbital character at finite wavelength. These predictions can be probed in inelastic light scattering experiments.

Figure 1

Bilayer graphene in Bernal stacking. The hopping parameters of the Slonczewski-Weiss-McClure model (Ref. 60), conventionally denoted ${\gamma }_0$, ${\gamma }_1$, ${\gamma }_3$, and ${\gamma }_4$, are also indicated.

Figure 2

The overlap of the Landau orbitals with the “ideal” limit, ${\gamma }_3={\gamma }_4={\Delta }^{\prime }=0$, as the SWM parameters ${\gamma }_3,{\gamma }_4,{\Delta }^{\prime }$ are gradually tuned from zero to their literary values [Eqs. (7) to (8)] for the lowest-energy Landau levels. For the effect of ${\gamma }_3$ in the two-band model, see Ref. 66.

Figure 3

Magnetoexciton modes at the $\nu =-4$ integer quantum Hall effect in bilayer graphene. Only the optically relevant spin- and pseudospin-conserving modes are shown. In the $q\to 0$ limit, which is probed by purely optical experiments, the transitions in the infinite sequence at fixed $l_z=\left|n\right|-|n^{\prime }|$ may mix; at $q>0$, the excitations result from the mixing of all $l_z$ sequences. The modes at $\nu =+4$ are obtained by particle-hole conjugation.

Figure 4

The excitations of the integer quantum Hall state at $\left|\nu \right|=4$ and $B=10$ T. The mixing of Landau levels is truncated at $L=1$ and $M=7$. Solid lines show the 15-fold degenerate excitations, which include three optically relevant $S_z=P_z=0$ modes. Dashed lines show the spin and pseudospin singlets. Inset: spectra if Landau level mixing is neglected. Side panels: the weight of the definite $l_z$ projections in each curve in bottom-up order. The top row shows the spin and pseudospin singlets.

Figure 5

The excitations of the integer quantum Hall state at $\left|\nu \right|=8$ and $\left|\nu \right|=12$, at $B=10$ T.

Figure 6

Magnetoexciton modes at the $\nu =0$ quantum Hall ferromagnetic state in bilayer graphene. Only the optically relevant spin- and pseudospin-conserving modes are shown. In the $q\to 0$ limit, which is probed by photon absorption or electronic Raman, the transitions in the infinite sequence at fixed $l_z=\left|n\right|-|n^{\prime }|$ may mix; at $q>0$, the excitations result from the mixing of all $l_z$ sequences.

Figure 7

Imagine two transitions: (I) a particle in level $n^{\prime }$ is promoted to level $n$ of a valley-spin component (left) of which the $n=1$ Landau band is filled, and (II) $-n\to -n^{\prime }$ of a component (right) of which the $n=1$ band is empty. Particle-hole conjugation of (I) is the promotion of a hole from Landau level $-n^{\prime }$ to $-n$, i.e., (II); the excitation energies must be the same. The consequence is Eq. (27); the formal proof is straightforward.

Figure 8

(a) Excitation spectra of the quantum Hall ferromagnet at $\nu =0,\pm 2$ at $B=10$ T. Only the optically relevant $S_z=P_z=0$ modes are included. (b) The weight of the shown excitations on the definite $l_z$ subspaces for $\nu =0$ in bottom-up order. For degenerate curves, the weight is summed. The quantum numbers of the $q\to 0$ limit are indicated.

Figure 9

The spin- and pseudospin-conserving magnetoexciton modes at the $\nu =-2$ quantum Hall ferromagnetic state in bilayer graphene. The modes at $\nu =+2$ are obtained by particle-hole conjugation.

Figure 10

The projection of the excitations at $\nu =-2$ on the definite $l_z$ subspaces. For degenerate curves, the weight is summed. The spectrum at $\nu =\pm 2$ is identical to the one in the left panel of Fig. 8, the letters refer to the same curves. For $\nu =+2$ the sign of the $+l_z$ and $-l_z$ projections are interchanged w.r.t. $\nu =-2$.

Figure 11

The optically spin- and pseudospin-conserving magnetoexciton modes at the $\nu =-3$ quantum Hall ferromagnetic state in bilayer graphene. Notice that the $S\downarrow$, $A\uparrow$, and $A\downarrow$ transitions occur symmetrically, i.e., the same electron and hole Landau levels are allowed and the same self-energy cost is picked up from the exchange with the filled levels.

Figure 12

(a) Excitation spectrum of the quantum Hall ferromagnet in the $\nu =-3$ at $B=10$ T. Only the optically relevant $S_z=P_z=0$ modes are included. (b) The weight of the shown excitations on the definite $l_z$ subspaces in bottom-up order. For degenerate curves, the weight is summed.