Ising quantum Hall ferromagnetism in Landau levels $\left|N\right|\ge 1$ of bilayer graphene

A magnetic field applied perpendicularly to the chiral two-dimensional electron gas (C2DEG) in a Bernal-stacked bilayer graphene quantizes the kinetic energy into a discrete set of Landau levels $N=0,\pm 1,\pm 2,....$ While Landau level $N=0$ is eightfold degenerate, higher Landau levels $\left(\left|N\right|\ge 1\right)$ are fourfold degenerate when spin and valley degrees of freedom are counted. In this work, the Hartree-Fock approximation is used to study the phase diagram of the C2DEG at integer fillings $\tilde{\nu }=1,2,3$ of these higher Landau levels. At these filling factors, the C2DEG is a valley or spin Ising quantum Hall ferromagnet. At odd fillings, the C2DEG is spin polarized and has all its electrons in one valley or the other. There is no intervalley coherence, in contrast with most of the ground states in Landau level $N=0.$ At even filling $\tilde{\nu }=2,$ the C2DEG is either fully spin polarized with electrons occupying both valleys or spin unpolarized with electrons occupying one of the two valleys. A finite electric field (or bias) applied perpendicularly to the plane of the C2DEG induces a series of first-order phase transitions between these different ground states. The transport gap or its slope is discontinuous at the bias where a transition occurs. Such discontinuity may result in a change in the transport properties of the C2DEG at that bias.

Figure 1

Labeling of the energy levels for one valley and one spin component at zero bias and for ${\gamma }_4={\delta }_0={\Delta }_B=0$. The vertical position of each line does not reflect the actual energy of each level.

Figure 2

(a) Dispersion of the Landau levels $N=0,1,2$ with bias ${\Delta }_B$ for $B=10$ T; (b) part of the same energy spectrum for level $N=1$ showing the crossing between the sublevels $(K_{+},+)$ and $(K_{-},-).$

Figure 3

(a) Coefficients $|c_{\xi ,i}{|}^2$ and (b) projectors $n_{1,\pm },n_{2,\pm },n_1,n_2$ as functions of bias for $B=10$ T and Landau level $N=1$.

Figure 4

Effective interactions (a) ${\Lambda }_z$ and (b) $J_z,J_{\perp }$ of the Ising quantum Hall valley ferromagnet as a function of the applied bias for $\tilde{\nu }=1$ and $N=1,2,3$ at $B=16.5$ T, $\kappa =2.5.$ The interactions $X^{++}$ and $X^{--}$ are also plotted for $N=1.$ In (a) the left axis is for $N=1$ and the right axis for $N=2,3.$

Figure 5

Critical bias ${\Delta }_B^{\left(c\right)}$ as a function of the magnetic field in Landau levels $N=1,2$ at $\tilde{\nu }=1$ for $\kappa =2.5$ and $\kappa =5.$ The region below (above) each curve is valley polarized with $P_z=+\frac{1}{2}(-\frac{1}{2}).$ Biases above ${\Delta }_B^{\left(c\right)}=0.10$ eV are outside the limit of validity of the model.

Figure 6

(a) Hartree-Fock gap ${\Delta }_{eh}$ (left axis, black curve) and charge imbalance ${\rho }_1-{\rho }_2$ (right axis, blue dashed curve) plotted as functions of the applied bias for $N=1,B=16.5$ T, $\tilde{\nu }=1,$ and $\kappa =2.5$. Both functions are discontinuous at the transition indicated by the dash-dotted line. (b) Hartree-Fock gap vs bias for $N=-1,-2,-3$ at $B=10$ T and $\kappa =2.5.$

Figure 7

Phase diagram in Landau level $N=1$ for filling factor $\tilde{\nu }=2$ and dielectric constant $\kappa =2.5.$ The red dashed line with the filled red circles is the noninteracting result for the critical bias.

Figure 8

Phase diagram in Landau level $N=2$ for filling factor $\tilde{\nu }=2$ and dielectric constant $\kappa =2.5.$ The red dashed line with the filled red circles is the noninteracting result for the critical bias.

Figure 9

Critical bias ${\Delta }_B^{\left(c\right)}$ (left axis) and critical electric field $E^{\left(c\right)}={\Delta }_B^{\left(c\right)}/ed$ (right axis) for the spin-polarized to spin-unpolarized phase transition with electrons in valley $K_{-}$ for filling factor $\tilde{\nu }=2$ and dielectric constant $\kappa =2.5$ in Landau levels $N=-1,-2,-3.$ The dashed lines give the noninteracting results for the critical bias ${\Delta }_B^{\left(c\right)}$ and electric field $E^{\left(c\right)}.$

Figure 10

Hartree-Fock gap ${\Delta }_{eh}$ (left axis, full line) and charge imbalance ${\rho }_1-{\rho }_2$ (right axis, dashed line) vs applied bias for $\tilde{\nu }=2,B=10$ T, $\kappa =2.5$, and $N=1.$ The dash-dotted line indicates the critical bias for the phase transition.

Figure 11

Hartree-Fock gap ${\Delta }_{eh}$ as a function of the applied bias for $\tilde{\nu }=2$ and $B=20,23,25$ T, $\kappa =2.5,$ and $N=1.$

Figure 12

Hartree-Fock gap ${\Delta }_{eh}$ as a function of the applied bias for $\tilde{\nu }=2,$ $B=10$ T, $\kappa =2.5,$ and $N=-1,-2,-3.$