Infrared Spectroscopy of Landau Levels of Graphene

We report infrared studies of the Landau level (LL) transitions in single layer graphene. Our specimens are density tunable and show in situ half-integer quantum Hall plateaus. Infrared transmission is measured in magnetic fields up to $B=18\mathrm{T}$ at selected LL fillings. Resonances between hole LLs and electron LLs, as well as resonances between hole and electron LLs, are resolved. Their transition energies are proportional to $\sqrt{B}$, and the deduced band velocity is $\tilde{c}\approx 1.1\times {10}^6\mathrm{m}/\mathrm{s}$. The lack of precise scaling between different LL transitions indicates considerable contributions of many-particle effects to the infrared transition energies.

Figure 1

Half-integer quantum Hall effect in graphene with plateaus at $R_{xy}=h/\nu e^2$ for filling factors $\nu =\pm 2,\pm 6,\pm 10$, at 4.2 K and 18 T. LL filling factors are indicated. Note that our devices are density tunable with $n_s/V_g=7.2\times {10}^{10}{\mathrm{cm}}^{-2}{\mathrm{V}}^{-1}$, where $n_s$ is the charge carrier density in graphene. Upper inset: Optical image of an $1100\mu {\mathrm{m}}^2$ graphene device. Lower inset: Schematic of the experimental setup.

Figure 2

Normalized IR absorption spectra of holes in graphene at three different magnetic fields, taken by dividing spectra taken at filling factors $\nu =-2$ and $\nu =-10$. Two LL resonances are denoted by $T_1$ and $T_2$. Residual spectral artifacts are associated with 60 Hz harmonics and with carriers in the Si substrate. Dashed purple lines are Lorentzian fits to the data. The inset shows a schematic LL ladder with allowed transitions indicated by arrows.

Figure 3

(a) Resonance energies vs $\sqrt{B}$, from holes (ratio of $\nu =-2$ and $\nu =-10$ data, Fig. 2) and electrons (ratio of $\nu =+2$ and $\nu =+10$, spectra not shown). Similar behavior has been observed in a second sample (squares). The solid line is a best $\sqrt{B}$ fit to the $T_1$ transition, yielding $\tilde{c}=(1.12\pm 0.02)\times {10}^6\mathrm{m}/\mathrm{s}$. The dashed line represents a scaling of the solid line by a factor of $(\sqrt{2}+1)$. (b) Half width at half maximum from Lorentzian fits of the $T_1$ transitions, as a function of $\sqrt{B}$.