Spin splitting in graphene studied by means of tilted magnetic-field experiments

We have measured the spin splitting in single-layer and bilayer graphene by means of tilted magnetic-field experiments. By applying the Lifshitz-Kosevich formula for the spin-induced decrease of the Shubnikov-de Haas amplitudes with increasing tilt angle, we directly determine the product between the carrier cyclotron mass $m^{*}$ and the effective $g$ factor $g^{*}$ as a function of the charge-carrier concentration. By using the cyclotron mass for a single-layer and a bilayer graphene, we find an enhanced $g$ factor $g^{*}=2.7\pm 0.2$ for both systems.

Figure 1

Shubnikov-de Haas oscillations in SLG (a) at $T=1.3$ K and in BLG(b) at $T=0.4$ K as a function of the carrier concentration for different total fields $B_{\text{tot}}$ or tilt angles $\theta$, respectively. When varying $\theta$, the total field $B_{\text{tot}}$ is adjusted such that the perpendicular field $B_n$ remains constant, i.e., $B_{\text{tot}}=B_n/\cos \theta$. The oscillation maxima are marked with the corresponding Landau-level numbers $N$. The inset schematically shows this tilting configuration.

Figure 2

Normalized oscillation amplitudes as a function of total field $B_{\text{tot}}$ at a constant perpendicular field $B_n$ in SLG (a) and BLG (b). Error bars represent standard least-squares-fitting errors in the determination of $A$. Solid lines are fits to Eq. (2) with $m^{*}{g}^{*}$ as a fit parameter.

Figure 3

Schematic representation of the density of states for a Landau level with an increasing total magnetic field $B_{\text{tot}}$ (from the bottom to the top) at a constant perpendicular component $B_n$ (a). (b) shows this scenario as measured experimentally for the $N=9,10$ maximum in SLG at a constant perpendicular magnetic field $B_n=5$ T.

Figure 4

Experimentally deduced $m^{*}{g}^{*}$ (open symbols), normalized to the free-electron mass $m_e$, as a function of charge-carrier concentration for SLG (a) and BLG (b). The individual data points were extracted from the total-field dependence of the SdH amplitudes corresponding to different Landau levels $N=2,...,10$ and represent measurements for a constant magnetic field $B_n=5$, 6, and 7 T for SLG and $B_n=8$ T for BLG. The error bars represent the standard least-squares-fitting errors, taking into account the error bars of $A$ (Fig. 2). The dashed lines in (a) represent the calculated behavior of $m^{*}{g}^{*}$ for different values of $g^{*}$, taking into account a 10$\%$ experimental uncertainty (shadowed areas). The crosses in (b) compare our data to the experimental cyclotron mass for BLG (Ref.17) multiplied by $g^{*}=2.5$. The inset shows the effective $g$ factor, extracted from the product $m^{*}{g}^{*}$ in the main panel and the known cyclotron mass $m^{*}$ in SLG, as a function of Landau-level index $N$.