Quantized Landau level spectrum and its density dependence in graphene

Scanning tunneling microscopy and spectroscopy in a magnetic field was used to study Landau quantization in graphene and its dependence on charge carrier density. Measurements were carried out on exfoliated graphene samples deposited on a chlorinated SiO${}_2$ thermal oxide, which allowed for the observation of the Landau level sequences characteristic of single-layer graphene while tuning the density through the Si backgate. Upon changing the carrier density, we find abrupt jumps in the Fermi level after each Landau level is filled. Moreover, at low doping levels, a marked increase in the Fermi velocity is observed, which is consistent with the logarithmic divergence expected due to the onset of many-body effects close to the Dirac point.

Figure 1

(a) Sketch of the experimental setup showing a graphene flake deposited on Si/SiO${}_2$, the STM tip above, and the backgate connected to the graphene. (b) STM topography image of a typical 300 nm $\times$ 300 nm graphene area. Tunneling conditions: $I_t=20\hspace{0.5em}\mathrm{pA},V_{\text{bias}}=190\hspace{0.5em}\mathrm{mV}$. (c) STM atomic resolution image on the graphene flake ($I_t=20\hspace{0.5em}\mathrm{pA},V_{\text{bias}}=300\hspace{0.5em}\mathrm{mV}$). (d) Profile line through the topography image [at the position indicated in (b)] showing the corrugation of graphene on SiO${}_2$ substrate.

Figure 2

(a) Topography image corresponding to the local density of states (LDOS) map in (b) $(I_t=20\hspace{0.5em}\mathrm{pA},V_{\text{bias}}=140\hspace{0.5em}\mathrm{mV})$. (b) LDOS map taken at $E=140$ meV showing doping inhomogeneity in graphene. (c) STS spectrum at $B=0$ T showing smearing of the Dirac point (indicated by $E_D$) due to disorder.

Figure 3

Landau levels in ungated graphene on SiO${}_2$ ($V_G=0$ V). (a) Scanning tunneling spectroscopy data in low fields $B=1$ –5.5 T together with $B=0$ T and $B=7$ T data for comparison. (b) Landau levels in higher fields $B=7$ T to $B=12$ T, clearly resolving LL with indices $N=-4,-3,-2,-1,0$. (c) Energy of the Landau levels as a function of $\text{sgn}(N)\sqrt{\left|N\right|B}$. From the linear dependence, specific to Dirac fermions, the Fermi velocity is extracted.

Figure 4

(a) Measured STS as function of gate voltage $V$${}_G$ (carrier density $n$). Each vertical line in the map is a spectrum at a particular gate. The bright stripes correspond to Landau Levels as indicated by $N=0,\pm 1,\pm 2$. (b) Simulation of the evolution of the LL spectrum with gate voltage, assuming Lorenzian line shapes of width $\approx$30 meV, for all levels and $v_F=(1.16\pm 0.02)\times 10$${}^6$ m/s. The latter was obtained from fitting the LLs at $V_G=0$ V and $C=100$ aF$\mu$m${}^{-2}$ for the capacitance between the graphene and the Si backgate. These data and those in Fig. 3 were taken in different parts of the sample, hence, the small variation in Fermi velocity. (c) Gate voltage dependence of the Fermi velocity. The Dirac point, at $V_G=35$ V, is marked. (d) Illustration of the effect of gating on the Fermi level.