Transport gap and hysteretic behavior of the Ising quantum Hall ferromagnets in $\left|N\right|>0$ Landau levels of bilayer graphene

The chiral two-dimensional electron gas in Landau levels $\left|N\right|>0$ of a Bernal-stacked graphene bilayer has a valley-pseudospin Ising quantum Hall ferromagnetic behavior at odd filling factors ${\nu }_N=1,3$ of these fourfold degenerate states. At zero interlayer electrical bias, the ground state at these fillings is spin polarized and electrons occupy one valley or the other while a finite electrical bias produces a series of valley pseudospin-flip transitions. In this work, we discuss the hysteretic behavior of the Ising quantum Hall ferromagnets. We compute the transport gap due to different excitations: bulk electron-hole pairs, electron-hole pairs confined to the coherent region of a valley-pseudospin domain wall, and spin or valley-pseudospin skyrmion-antiskyrmion pairs. We determine which of these excitations has the lowest energy at a given value of the Zeeman coupling, bias, and magnetic field.

Figure 1

(a) General form of the energy functional $e_{\mathrm{HF}}\left(\theta \right)$ with the two minima at $\theta =0,\pi$ and the maximum at ${\theta }^{*}.$ Energy per electron at the extremal values of $e_{\mathrm{HF}}\left(\theta \right)$ for (b) $N=-1,B=10\mathrm{T};$ (c) $N=1,B=21\mathrm{T}$.

Figure 2

Energy of the bulk $({\Delta }_{\mathrm{spin},\xi }$ and ${\Delta }_{\mathrm{valley},\xi })$ and confined $\left({\Delta }_{\mathrm{cf},-}\right)$ electron-hole excitations as a function of bias for (a) $N=-1,B=10$ T and (b) $N=1,B=21$ T.

Figure 3

Domain wall for ${\Delta }_B=0,N=-1,B=10$ T and hysteretic behavior. (a) Comparison of the sine-Gordon and Hartree-Fock solutions. (b) Band structure, $E_{\pm }\left(X\right)$ in the domain wall solution. The gap for a confined electron-hole excitation is indicated by ${\Delta }_{\mathrm{cf}}.$ (c) Hysteretic behavior of the transport gap around zero bias. The plain (dashed) arrows indicate the progression of the gap for a downward (upward) sweep in bias.

Figure 4

Excitation energy at zero bias of the (a) antiskyrmion and bulk quasihole; (b) skyrmion and bulk quasielectron as a function of the Zeeman coupling for Landau level $N=-1$ and magnetic field $B=10$ T.

Figure 5

Number of spin flips $\left(K_{\mathrm{spin}}\right)$ or valley pseudospin flips $\left(K_{\mathrm{pspin}}\right)$ as a function of the Zeeman coupling for the lowest-energy excitations in Fig. 4.

Figure 6

Excitation energy of (a) valley antiskyrmion and quasihole; (b) valley skyrmion and quasielectron for bias ${\Delta }_B=0.03e^2/\kappa \ell , B=10$ T, and $N=-1.$ The excitations are ${\left|e\right.〉}_1$ and ${\left|Sk\right.〉}_1$ for positive bias and ${\left|e\right.〉}_3$ and ${\left|Sk\right.〉}_3$ for negative bias.

Figure 7

Number of flipped valley pseudospins per skyrmion or antiskyrmion as a function of bias in Landau level $N=-1$ and magnetic field $B=10$ T.

Figure 8

Valley-pseudospin texture of a valley-pseudospin skyrmion in Landau level $N=-1$ and magnetic field $B=10$ T in the projected representation.

Figure 9

Energy with bias of bulk and confined electron-hole pairs and of valley-pseudospin skyrmion-antiskyrmion pairs. (a) $N=-1,B=10\mathrm{T}$; (b) $N=1,B=21$ T.