Landau levels and oscillator strength in a biased bilayer of graphene

We obtain analytical expressions for the eigenstates and the Landau level spectrum of biased graphene bilayers in a magnetic field. The calculations are performed in the context of a four-band continuum model and generalize previous approximate results. Solutions are presented for the spectrum as a function of interlayer coupling, the potential difference between the layers, and the magnetic field. The explicit expressions allow us to calculate the oscillator strength and the selection rules for electric dipole transitions between the Landau states. Some transitions are significantly shifted in energy relative to those in an unbiased bilayer and exhibit a very different magnetic field dependence.

Figure 1

(Color online) Low-energy electronic dispersion of graphene bilayers for $B=0$ for different values of the gap, namely, $\Delta U=0$ (black solid line), $\Delta U=25\mathrm{meV}$ (red dashed line), $\Delta U=100\mathrm{meV}$ (blue dotted), and $\Delta U=200\mathrm{meV}$ (green dot-dashed line). In all cases, $U_0=0$.

Figure 2

(Color online) Landau levels $(n=0–5)$ in an unbiased graphene bilayer as a function of the magnetic field. The solid, red lines result from a numerical solution of Eq. (25) and the dashed, blue ones are the approximate results from Eq. (27).

Figure 3

(Color online) (a) Positive-energy Landau levels in a biased graphene bilayer as a function of the magnetic field $B$. (b) As in (a) for negative-energy levels. In both cases, $U_1=-U_2=50\mathrm{meV}$ and $n=0$ (solid line), $n=1$ (dashed lines), $n=2$ (dotted), $n=3$ (long dashed), $n=4$ (dash dotted), and $n=5$ (dot-dot-dashed lines). The rectangular areas are enlarged in the insets.

Figure 4

Landau levels in an unbiased graphene bilayer, for $B=5\mathrm{T}$, as a function of the interlayer coupling parameter $t$.

Figure 5

(Color online) Landau levels in a graphene bilayer as a function of the difference $\Delta U$ in potential between the layers for $B=5\mathrm{T}$: $n=0$ (black solid line), $n=1$ (red dashed lines), $n=2$ (blue dotted lines), $n=3$ (green long-dashed lines), $n=4$ (orange dash-dotted lines), and $n=5$ (magenta dot-dot-dashed lines).

Figure 6

(Color online) (a) Oscillator strengths for electric dipole transitions in a biased bilayer of graphene $(\Delta U=100\mathrm{meV})$ for $B=10\mathrm{T}$. (b) As in (a) for an unbiased $(\Delta U=0\mathrm{meV})$ bilayer.

Figure 7

(Color online) (a) Transition energies in a biased $(\Delta U=100\mathrm{meV})$ graphene bilayer, as a function of the magnetic field $B$, for the dipole-allowed transitions: (0, 1) (black dashed line), (1, 2) (blue dotted line), $(-1,-2)$ (black solid line), and $(-1,2)$ (red solid line). (b) Oscillator strengths vs field $B$ for the transitions described in (a).

Figure 8

(Color online) Transition energies as a function of $\Delta U$ for the (1,2) (blue solid line) and $(-1,-2)$ (red dashed line) transitions for $B=20\mathrm{T}$, and (1,2) (green solid line) and $(-1,-2)$ (black dotted line) for $B=10\mathrm{T}$.

Figure 9

(Color online) Oscillator strengths for dipole-allowed transitions in a graphene bilayer as a function of the interlayer potential difference $\Delta U$ for the (0, 1) (black dashed line), (1, 2) (blue dotted line), $(-1,-2)$ (black solid), and $(-1,2)$ (red solid line) transitions at $B=5\mathrm{T}$.

Figure 10

(Color online) (a) Energies of the (1, 2) and $(-1,-2)$ (upper curves of each doublet) transitions as a function of magnetic field for $\Delta U=0$ (black solid line), $\Delta U=25\mathrm{meV}$ (red dashed lines), $\Delta U=50\mathrm{meV}$ (blue dotted lines), $\Delta U=100\mathrm{meV}$ (green dot-dashed lines), and $\Delta U=200\mathrm{meV}$ (magenta dot-dot-dashed lines). (b) The oscillator strengths for the transitions shown in (a) as a function of magnetic field. The results below 0.92 correspond to the (1,2) transitions.