Bound states in inhomogeneous magnetic field in graphene: Semiclassical approach

We derive semiclassical quantization equations for graphene monolayer and bilayer systems where the excitations are confined by the applied inhomogeneous magnetic field. The importance of a semiclassical phase, a consequence of the spinor nature of the excitations, is pointed out. The semiclassical eigenenergies show good agreement with the results of quantum-mechanical calculations based on the Dirac equation of graphene and with numerical tight-binding calculations.

Figure 1

In (a) and (b) the applied perpendicular magnetic field $\mathbf{B}$ is zero in the center region of width $2W$. In the case of (a), in the left and right regions the magnetic fields point in the same directions, while in the case of (b), in the opposite directions. The magnitude $B$ of the magnetic field is the same in both left and right regions. In the case of Fig. 1c a circular nonmagnetic region of radius $R$ is considered in graphene monolayer, whereas outside this region there is perpendicularly applied magnetic field of magnitude $B$.

Figure 2

Results of exact quantum calculations (solid lines) and the semiclassical approximation given by Eqs. (15, 16) (circles) as a function of $k_y$ (in units of $l_B$) for graphene monolayer. The dashed lines indicate $\left|\tilde{X}\right|={\tilde{E}}_{ml}$ (see text).

Figure 3

Results of exact quantum calculations (solid lines) and the semiclassical approximation given by Eqs. (17, 18) (circles) as a function of $k_y$ (in units of $l_B$) for graphene bilayer. The dashed lines indicate ${\tilde{X}}^2/2={\tilde{E}}_{bl}$ (see text).

Figure 4

Results of exact quantum calculations (solid lines) and the semiclassical approximation given by Eq. (19) (circles) as a function of $k_y$ (in units of $l_B$) for graphene monolayer. The dashed line indicates $\tilde{X}={\tilde{E}}_{ml}$ (see text).

Figure 5

Results of TB calculations (solid lines) and a semiclassical approximation (circles) for graphene bilayer as a function of $k_y$ (in units of $l_B$). The semiclassical approximation can be obtained from Eq. (19) by a transformation described in the main text. The dashed line indicates ${\tilde{X}}^2/2={\tilde{E}}_{bl}$ (see text).

Figure 6

(a) Results of exact quantum calculations (solid lines) for the $m=-2$ energy bands as a function of the missing flux $\delta$. The results of the semiclassical quantization $(\ast )$ are obtained from Eq. (23) for $\tilde{E}<\frac{\left|m\right|}{\sqrt{2\delta }}$ and $\tilde{m}<0$. For $\tilde{E}>\frac{\left|m\right|}{\sqrt{2\delta }}$ the circles $(○)$ show the semiclassical results calculated using Eqs. (24, 25). The dashed line shows the $\tilde{E}=\frac{\left|m\right|}{\sqrt{2\delta }}$ function which separates the cases (i) and (ii) detailed in the main text; (b) shows a cartoon of a classical orbit in the parameter range $\tilde{E}<\frac{\left|m\right|}{\sqrt{2\delta }}$ and $\tilde{m}<0$.

Figure 7

(a) Results of exact quantum calculations (solid lines) for the $m=2$ energy bands as a function of the missing flux $\delta$. The results of the semiclassical quantization $(\ast )$ are obtained from Eq. (23) for $\tilde{E}<\frac{\left|m\right|}{\sqrt{2\delta }}$ and $\tilde{m}>0$. For $\tilde{E}>\frac{\left|m\right|}{\sqrt{2\delta }}$ the circles $(○)$ show the semiclassical results calculated using Eqs. (24, 25). The dashed line shows the $\tilde{E}=\frac{\left|m\right|}{\sqrt{2\delta }}$ function which separates the cases (i) and (ii) detailed in the main text; (b) shows a cartoon of a classical orbit in the parameter range $\tilde{E}<\frac{\left|m\right|}{\sqrt{2\delta }}$ and $\tilde{m}>0$.