Resonance Microwave Measurements of an Intrinsic Spin-Orbit Coupling Gap in Graphene: A Possible Indication of a Topological State

In 2005, Kane and Mele [Phys. Rev. Lett. 95, 226801 (2005)] predicted that at sufficiently low energy, graphene exhibits a topological state of matter with an energy gap generated by the atomic spin-orbit interaction. However, this intrinsic gap has not been measured to this date. In this Letter, we exploit the chirality of the low-energy states to resolve this gap. We probe the spin states experimentally by employing low temperature microwave excitation in a resistively detected electron-spin resonance on graphene. The structure of the topological bands is reflected in our transport experiments, where our numerical models allow us to identify the resonance signatures. We determine the intrinsic spin-orbit bulk gap to be exactly $42.2\mu \mathrm{eV}$. Electron-spin resonance experiments can reveal the competition between the intrinsic spin-orbit coupling and classical Zeeman energy that arises at low magnetic fields and demonstrate that graphene remains to be a material with surprising properties.

Figure 1

(a) Band structure of a honeycomb lattice terminated on a zigzag edge with periodic boundary conditions along the armchair direction, colored according to their chirality, $⟨{\widehat{h}}_k⟩$: Green (magenta) denotes positive (negative) chirality. The states crossing the gap are flat on this energy scale (high DOS). (b) Magnified dispersion relation near the Fermi level of the first Brillouin zone, showing the edge states in detail. The bands are now colored according to their helicity $⟨{\widehat{h}}_k{⟩}_{\mathrm{edge}}$, with black denoting bulk character. (c),(d) Same as in (b) for the $E1$ bands, with $B={\mathrm{\Delta }}_I/2$ and $B=2{\mathrm{\Delta }}_I$, respectively, in units of $g{\mu }_B$. The spin on each band is indicated with a color matching arrow.

Figure 2

Schematics of the measurement setup with monolayer graphene patterned into a Hall-bar structure. A nearby loop antenna excites the system (not to scale).

Figure 3

(a) Individual measurements at $T=4.2\mathrm{K}$ for various frequencies with constant gate voltage $\mathrm{\Delta }{V}_{\mathrm{CNP}}=V_g-V_{\mathrm{CNP}}=4\mathrm{V}$. Two resonances which are symmetric in $B$ exhibit a linear dependence on $\nu$ (dashed lines). The feature at zero field stems from the weak localization in the sample. The data have been shifted and scaled for clarity. (b) Derivative data of all measurements recorded for $0.5\mathrm{GHz}\le \nu \le 40\mathrm{GHz}$. The resonance signal is present over a wide range of frequencies except for $\nu \lesssim 11\mathrm{GHz}$. The upper $V$ feature intercepts the frequency axis at ${\nu }_1=10.2\mathrm{GHz}$, while the lower feature interpolates to ${\nu }_2=0$. This difference corresponds to an energy of $\mathrm{\Delta }E=42.2\mu \mathrm{eV}$.

Figure 4

Individual measurements at $T=1.4\mathrm{K}$ for various gate voltages $\mathrm{\Delta }{V}_{\mathrm{CNP}}$ with constant frequency $\nu =23\mathrm{GHz}$, with red (blue) indicating electron (hole) character of the main charge carriers. The two symmetric resonance peaks are evident at $|B_1|=0.48\mathrm{T}$ and $|B_2|=0.85\mathrm{T}$ (dashed lines), and their positions are invariant to gate voltage, i.e., to charge carrier density and type. The data have been shifted and scaled for clarity.