Figure 1

Lattice structure and Brillouin zone of monolayer graphene. Left: Hexagonal lattice of graphene, with the next nearest neighbor ${\mathit{\delta }}_i$ and the primitive ${\mathit{a}}_i$ vectors depicted. The area of the primitive cell is $A_c=3\sqrt{3}{a}_0^2/2\simeq 5.1$ Å${}^2$, and $a_0\simeq 1.4$ Å. Right: Brillouin zone of graphene, with the Dirac points $\mathit{K}$ and ${\mathit{K}}^{\prime }$ indicated. Close to these points, the dispersion of graphene is conical and the density of states is proportional to the absolute value of the energy.Reuse & Permissions

Figure 2

Allowed interband transitions (vertical arrows) in graphene; a photon of energy $\hslash \omega$ produces an excitation from the lower to the upper Dirac, as long as $\hslash \omega >2\mu$. The transitions conserve $\mathbf{k}$ and hence are said to be “vertical.” For $\hslash \omega \le 2\mu ,$ Pauli blocking forbids any (interband) transition. In practice, due to disorder (impurities, etc.), the interband conductivity can be nonzero even for $\hslash \omega \le 2\mu$.Reuse & Permissions

Figure 3

Schematic of electronic transitions contributing to ${\sigma }_{xx}\left(\omega \right)$ of doped graphene in a magnetic field. In this example, $E_F\ge E_1$, and thus the last occupied LL, $n=N_F\ge 1$, belongs to the conduction band. Two types of transitions take place: (i) interband transitions, connecting LLs from the lower cone (valence band) with LLs in the upper cone (conduction band); and (ii) intraband transitions within the upper cone. Intraband transitions are limited to adjacent LLs: $N_F\to N_F+1$. The figure shows the following interband transitions: (a) the pair $-N_F\to N_F+1$ and $-N_F-1\to N_F$, whose energy difference is $E_{N+1}+E_N$ (the lowest interband energy; note that transitions $-N_F-1\to N_F$ are forbidden because $n=N_F$ is occupied); and (b) the pair $-N_F-1\to N_F+2$ and $-N_F-2\to N_F+1$. The respective energy difference is $E_{N+1}+E_{N+2}$ (the second lowest interband energy difference), and in this case both transitions take place. Transitions with higher energy differences are not represented.Reuse & Permissions

Figure 4

Longitudinal magneto-optical conductivity as a function of the photon energy for a field of 7 T, zero chemical potential, $T=17$ K, and $\Gamma =6.8$ meV ($\sim$79 K). The horizontal dashed-dot (black) line marks the graphene's universal ac-conductivity background [Eq. (1)].Reuse & Permissions

Figure 5

Longitudinal conductivity as a function of photon energy for $E_F=0.2$ eV. Other parameters as in Fig. 4. The solid horizontal (black) line shows graphene's universal ac-conductivity background [Eq. (1)].Reuse & Permissions

Figure 6

Schematic of electronic transitions contributing to the Hall conductivity of doped graphene in a magnetic field. Contrary to the longitudinal conductivity (Fig. 3), symmetry implies that only interband transitions involving the smallest energy difference, $\hslash \Delta {\Omega }_{N_F}=E_{N_F}+E_{N_F+1}$, contribute to ${\sigma }_{xy}$. The remaining interband transitions ($\Delta {\Omega }_n$, with $n>N_F$) come in pairs whose contribution to the Hall current mutually cancel as explained in the text: an example of a pair of interband transitions that cancel is shown in zigzag arrows. Note: The schematic picture is strictly adequate for $N_F\ge 1$; the case of $N_F=0$ admits a single type of electronic transition, namely, $n=0\to n=1$.Reuse & Permissions

Figure 7

Hall conductivity as a function of photon energy for $E_F=0$ (top) and $E_F=0.2$ eV (bottom). In both plots $T=0$ (other parameters as in Fig. 4). At zero Fermi energy (top), ${\sigma }_{xy}\left(\omega \right)$ originates in a single type of interband transition, centered at $\hslash \omega \approx E_1\simeq 96$ meV, and therefore cannot be described by a semiclassical treatment [Eq. (58)]. When $E_F=0.2$ eV (bottom), the first four LLs are fulfilled, which results in a classical intraband contribution [Eq. (57)], centered at $\hslash \omega \simeq E_5-E_4\approx 23$ meV, and a single interband transition [Eq. (58)] centered at $\hslash \omega \simeq E_5+E_4\approx 0.4$ eV $\simeq 2E_F$. The latter is shown in the inset.Reuse & Permissions

Figure 8

The dc Hall conductivity as a function of the Fermi energy. The parameters are $T=17$ K and $\Gamma =0.68$ meV. The plateaux show Hall quantization values according to the theoretical prediction for massless Dirac fermions [Eq. (64)].Reuse & Permissions

Figure 9

The real part of the longitudinal conductivity is plotted as a function of the photon energy for $E_F=0.1$ eV (left) and $E_F=0.15$ eV (right). In these plots, $B=7$ T and $\Gamma =6.8$ meV. The dashed (red) line represents the semiclassical result [Eq. (73)] and the solid (blue) line represents the EOM quantum solution [Eq. (42)]. Note that, in the right panel, there is no interband peak $n=-1\to m=2$, at $E_F\approx 230$ meV, and the peak at $\hslash \omega \approx 300$ meV loses half of its intensity because the $n=-3\to m=2$ transitions get blocked when the Fermi energy crosses the LL with $n=2$.Reuse & Permissions

Figure 10

Schematic of the Faraday effect: an electromagnetic wave polarized in the $xy$ plane (transverse magnetic mode) and traveling in the positive $z$ direction passes through a graphene film subjected to a transverse magnetic field $B$. In this case, graphene is adhered to a substrate (typically SiO${}_2$), but the experiment can also be made with suspended graphene. The transmitted field sees its plane of polarization rotated by an angle ${\theta }_F$ and acquires a certain degree of ellipticity.Reuse & Permissions

Figure 11

Faraday rotation angle (in degrees), normalized transmittance, and ellipticity of electromagnetic radiation passing through graphene subjected to a perpendicular magnetic field. The graphene sample is assumed to have a finite electronic density, $E_F=0.3$ eV, and to be on top of SiO${}_2$ (${\epsilon }_r=3.9$). Top six panels: Simulation of ${\theta }_F$, $T\left(B\right)/T\left(0\right)$, and $\delta$, considering a broadening of $\Gamma =7$ meV. Bottom six panels: Simulation of the same quantities as above for $\Gamma =3.7$ meV. In all panels, dashed lines correspond to approximate calculations, as given by Eqs. (89) and (91), and $T=17$ K.Reuse & Permissions

Figure 12

Faraday effect in doped graphene. Top: Faraday rotation angle (in degrees) when graphene is grown on silicon carbide. Fit to the experimental values of ${\theta }_F$, at a magnetic field of $B=7$ T (left) and $B=3$ T (right), using the semiclassical approach [dashed (green) line] and the full quantum calculation [solid (red) line]. Parameters are $E_F=0.3$ eV, $\Gamma =10.5$ meV, $T=6$ K, and ${\epsilon }_r=4.4$. Bottom: Theoretical optical conductivity [Eqs. (42) and (55)] for the same parameters used to fit the experimental data: $B=7$ T (left) and $B=3$ T (right).Reuse & Permissions

Figure 13

Low electronic density limit. Top: Faraday rotation angle (in degrees) for free-standing graphene (${\epsilon }_r=1$) for different LL occupations: from left to right, $N_F=0$, 1, and 2. The magnetic field intensity is $B=7$ T, $\Gamma =10.5$ meV, and $T=0$. Adding a dielectric substrate to graphene decreases the maximum amount of Faraday rotation that is achievable, without introducing major qualitative changes [see Eq. (89)]. Bottom: Real part of the quantum conductivity tensor for the Fermi energies considered in the top panel.Reuse & Permissions

Figure 14

Quantization of the Faraday effect in graphene. Top: Faraday rotation angle (in degrees) for free-standing graphene as a function of the Fermi energy at a magnetic field of $B=7$ T for $\hslash \omega =10$ meV (left) and $\hslash \omega =50$ meV (right). The respective semiclassical result is plotted by the dashed lines. Other parameters: $\Gamma =10.5$ meV and $T=12$ K. Bottom: Same as the top panel but with $\Gamma =2$ meV.Reuse & Permissions

Figure 15

Faraday rotation angle (in degrees) as a function of the magnetic field for $E_F=0.30$ eV (left) and $E_F=0.05$ eV (right). In each panel two photon energies are represented—$\hslash \omega =10$ meV [solid (blue) line] and $\hslash \omega =30$ meV [dashed–double-dotted (red) line]—with the respective semiclassical counterparts shown by dashed lines. Other parameters as in the top panel in Fig. 14.Reuse & Permissions

Figure 16

Schematic of the graphene-optical cavity system: linearly polarized light shines into an optical cavity with graphene placed at the center. The field inside the cavity perceives graphene as an extra boundary and hence the two halves of the cavity operate as independent cavities of effective size $L/2$. Matching the light frequency $\hslash \omega$ with a resonant frequency of the cavity $\omega =n\pi c/L$ ($n\in \mathbb{N}$) traps photons inside the cavity for several round trips. As a consequence, Faraday rotation accumulates due to multiple passages through graphene, leading to an output field with a large Faraday rotation.Reuse & Permissions

Figure 17

Faraday rotation angle of a cavity-graphene system in the semiclassical and quantum regimes. Top left: ${\theta }_F$ as a function of the photon energy for a cavity-graphene system in a magnetic field of 7 T. The Fermi energy reads $E_F=0.3$ eV and the cavity mirrors have $r=0.99$. Other parameters: $\Gamma =10.5$ meV and $T=12$ K. Top right: ${\theta }_F$ versus the reflection amplitude $r$ for $\hslash \omega =19$ meV. Bottom: In the left (right) panel, the Fermi energy reads $E_F=0.05$ eV ($E_F=0.1$ eV), which corresponds to an LL occupation of $N_F=0$ ($N_F=1$). The orange dashed line shows ${\theta }_F$ as obtained with the semiclassical conductivity tensor.Reuse & Permissions

Figure 18

Faraday rotation boost in the infrared and visible ranges. Left: Faraday rotation angle versus photon energy of a cavity-graphene system with $E_F=0.85$ eV (top) and $E_F=0.3$ eV (bottom)]. Right: Transmissivity of a cavity-graphene system for the same parameters considered at the left. Inset: Transmissivity of intrinsic graphene for the same parameters. Other parameters as in Fig. 17. In order to obtain a measurable Faraday rotation at $\hslash \omega \approx 1.7$ eV [solid (red) line] [$\hslash \omega \approx 0.5$ eV (infrared)], it is necessary to tune the intraband resonance according to $E_F\simeq \hslash \omega /2$.Reuse & Permissions

Figure 19

Schematic of an optical system consisting of an array of interfaces separated by different types of dielectric media. An electromagnetic wave, ${\mathit{E}}^{\mathrm{in}}={\mathit{E}}_1^{+}$, coming from a medium with dielectric permittivity ${\epsilon }_1$ interacts with an interface ${\alpha }_{12}$. As a result, it is partially reflected and partially transmitted into the medium ${\epsilon }_2$. Equivalent events take place at the remaining interfaces. The vectors with superscript $+$($-$) denote the component of the electric field traveling in the positive (negative) direction of $z$. A uniform static magnetic field $\mathbf{B}=B{\mathbf{e}}_{\mathbf{y}}$ is assumed.Reuse & Permissions