Fig. 3: Resolving the energies of the four isospin components of the graphene zero Landau level with Kelvin probe spectroscopy.

a Kelvin probe measurements varying the sample bias and simultaneously gates G1 and G2 for measurements made outside the Hall bar area. A staircase of plateaus shows various Landau levels occurring at different chemical potentials for various magnetic fields. b Chemical potential vs. filling factor given by the data in (a) collapsed onto a universal curve by scaling the sample bias by the graphene Landau level energy field dependence along the vertical axis, \({E}_{N}\propto {\!\,}^\surd B\), and by the \({B}^{-1}\) along the horizontal axis to give a density/filling factor axis. Each Landau level is observed by a plateau in the scaled chemical potential. Notice the zero Landau level at zero chemical potential consists of four separate small plateaus indicating the lifting of the fourfold degeneracy. c The Landau level density of states calculated using the expression in the main text with \(B=1\,\text{T}\) and \({v}_{F}=1.13\times {10}^{6}\text{m}/\text{s}\) to fit the locations of the plateaus in (b). d Blow-up of the large up and down excursion in chemical potential at \(\nu =-1\) and \(B=15\,\text{T}\) from (e). e Blow-up of the chemical potential of the zeroth Landau level at \(B=15\,\text{T}\) from (b) showing four individual chemical potential plateaus, labeled \({\varepsilon }_{i}\), separated by large up/down excursions at the incompressible filling factors, \(\nu =0,\pm 1\). The red dashed lines indicate the differences in chemical potential \(\triangle E=({\varepsilon }_{2}-{\varepsilon }_{1}),({\varepsilon }_{3}-{\varepsilon }_{2}),{\rm{and}}\,({\varepsilon }_{4}-{\varepsilon }_{3})\). f Energy differences extracted from the chemical potential plateaus in (e) for the \(\nu =0,({\varepsilon }_{3}-{\varepsilon }_{2})\) (red circles) and \(\nu =-1,({\varepsilon }_{2}-{\varepsilon }_{1}){\rm{and}}\) \(\nu =+1,({\varepsilon }_{4}-{\varepsilon }_{3})\) (orange triangles and green squares) filling factors. The values are averaged chemical potential difference values from \(\nu -0.75\) to \(\nu -0.25\) of each integer \(\nu\), and the error bars correspond to one standard deviation. The solid black line shows the Zeeman energy, \(g{\mu }_{B}B\), with \(g=2\). The solid red line is a fit for \(\nu =0\) data values to \(\sqrt{B}\) for values \(\ge 8\,\text{T}\), and the blue line is a linear fit for B values \(\le 8\,\text{T}\). AFM settings: 5.8 nm oscillation amplitude, \(\triangle f=-450\,{\rm{mHz}}\), 5 Hz bias modulation, except a 1 Hz bias modulation was used for 4 T and 5 T data.