Quantum Hall ferromagnetic phases in the Landau level $N=0$ of a graphene bilayer

In a Bernal-stacked graphene bilayer, an electronic state in Landau level $N=0$ is described by its guiding-center index $X$ (in the Landau gauge) and by its valley, spin, and orbital indices $\xi =\pm K,\sigma =\pm 1,$ and $n=0,1.$ When Coulomb interaction is taken into account, the chiral two-dimensional electron gas (C2DEG) in this system can support a variety of quantum Hall ferromagnetic ground states where the spins and/or valley pseudospins and/or orbital pseudospins collectively align in space. In this work, we give a comprehensive account of the phase diagram of the C2DEG at integer filling factors $\nu \in [-3,3]$ in Landau level $N=0$ when an electrical potential difference ${\Delta }_B$ between the two layers is varied. We consider states with or without layer, spin, or orbital coherence. For each phase, we discuss the behavior of the transport gap as a function of ${\Delta }_B,$ the spectrum of collective excitations, and the optical absorption due to orbital pseudospin-wave modes. We also study the effect of an external in-plane electric field on a coherent state that has both valley and spin coherence and show that it is possible, in such a state, to control the spin polarization by varying the strength of the external in-plane electric field.

Figure 1

Crystal structure of a Bernal-stacked graphene bilayer.

Figure 2

Brillouin zone of the triangular Bravais lattice and definition of the valleys indices $K_{\pm }.$

Figure 3

Comparison between the complete four-band model (symbols) and the approximate two-band model (lines) for the electronic dispersion in Landau levels $N=-2,-1,0,1,2$ and valley $K_{-}.$

Figure 4

Ordering of the four levels of a given spin in Landau level $N=0$ at finite bias ${\Delta }_B$.

Figure 5

Phase diagram of the C2DEG in Landau level $N=0$ at $B=10$ T and for $\kappa =5$ as a function of the transverse electric field between the layers for integer filling factors $\nu \in [-3,3]$ (from top to bottom). A filled (red) circle represents a filled state, while a filled (blue) ellipse indicates a coherent superposition of two levels. The numbering of the levels is indicated in the inset at the top right of the figure. The $E^{\left(i\right)\prime }$s indicate the critical perpendicular electric field (in mV/nm) required for the transition between two phases. Also indicated for each phase are the spin polarization $S_z,$ the number of Goldstone mode (G), of modes gapped at the Zeeman energy (Z), and the number of peaks in the optical absorption spectrum (A). The mention “$+$ nonuniform states” for the $O_1$ and $O_3$ phases signals that this portion of the phase diagram is further subdivided into uniform and nonuniform states as indicated in Eq. (109).

Figure 6

Variation of the populations and interlayer coherence with the transverse electric field in phase $L_{-1}.$

Figure 7

Variation of the populations interlayer coherence with the transverse electric field in phase $O_3.$

Figure 8

Variation of the transport gap with the perpendicular electric field in the different phases of the C2DEG in Landau level $N=0.$ The kink in the behavior of $I_{\pm 2}$ are due to level crossings. A value of $\kappa =5$ is assumed for the dielectric constant and $B=10$ T.

Figure 9

Dispersion of the collective modes in the coherent phases. The (blue) arrows point to the modes which are active in optical absorption experiments. All modes are evaluated at $B=10$ T with $\kappa =5.$ The bias in units of $e^2/\kappa \ell =35.6$ meV (corresponding to a frequency $\nu =e^2/h\kappa \ell =8.6\times {10}^{12}$ Hz) is, respectively, ${\Delta }_B=0.0005$ for $L_{-3},L_1;$ ${\Delta }_B=0.001$ for $L_3,L_{-1};{\Delta }_B=0.002$ for $L_{-2};{\Delta }_B=1.26$ for $O_3;$ ${\Delta }_B=1.115$ for $S{L}_{-1};{\Delta }_B=0.14$ for $S{L}_1$; and ${\Delta }_B=0.126$ for $S{L}_0.$ In $S{L}_0$, the middle line in each group of three dispersive curves contains two modes which are very close in energy.

Figure 10

Dispersion of the collective modes in the incoherent phases for $B=10$ T and $\kappa =5.$ The bias is in units of $e^2/\kappa \ell =35.6$ meV corresponding to a frequency $\nu =e^2/h\kappa \ell =8.6\times {10}^{12}$ Hz. The (blue) arrows point to the modes which are active in optical absorption experiments. The bias in units of $e^2/\kappa \ell$ is, respectively, ${\Delta }_B=0.01$ for $I_0,I_{\pm 1},I_{\pm 2},I_{\pm 3}$ and ${\Delta }_B=1.0$ for $I_0^{*},I_{\pm 1}^{*}.$ For $I_{\pm 3}$, $I_{\pm 2},I_{\pm 1}$, only the first four, three, and six modes, respectively, are shown. For $I_0$, each of the four branches contains four modes which are close in energy. In $I_0^{*}$ the middle line in each group of three curves contains two modes.

Figure 11

Electromagnetic absorption for $B=10$ T and $\kappa =5$ in phases (a) $L_{-3}$ (${\Delta }_B=0.018$ meV)$,$ $I_3$ (${\Delta }_B=0.36$ meV), and $L_{-2}$ (${\Delta }_B=0.07$ meV); (b) $S{L}_0$ (${\Delta }_B=4.48$ meV) and $S{L}_{-1}$ (${\Delta }_B=4.09$ meV)$.$

Figure 12

Variation of the spin and layer polarization and of the orbital coherence with an applied in-plane electric field in phase $SO{L}_{-1}.$ The bias ${\Delta }_B$ has been taken near the middle of the $S{L}_{-1}$ phase where $\langle S_z\rangle =\langle P_z\rangle \approx 1.$ Parameters are $B=10$ T and $\kappa =5.$

Figure 13

Electromagnetic absorption in the $SO{L}_{-1}$ phase for two different polarizations of the electromagnetic field.