Graphene is theoretically predicted to have a zero-energy Landau level, the energy width of which depends on the types of disorder; the energy width is ideally nearly zero in the presence of typical hopping disorder such as ripples, owing to protected chiral symmetry associated with graphene sublattice-symmetry. Previously, we developed transport energy-spectroscopy techniques relaying on densities of interface states involved in epitaxial graphene device [1]. Using the technique, here we report temperature dependences of the energy widths of the extended states of the zero-energy and first excited Landau levels, from which we can also deduce exponents corresponding to the critical exponents used in the quantum Hall plateau-plateau transition. The energy widths obtained from the spectroscopy technique are also compared with the energy widths deduced from the activation gap measurements [2].

Figure 1(a) shows longitudinal resistance as a function of gate voltage (*V*_{g}) and magnetic field (*B*). Due to the presence of the interface states in gate insulator or in SiC underneath the graphene (Fig. 1(b)), trajectories of the longitudinal-resistance peaks become parabolic, reflecting the unequally-spaced graphene Landau levels. Such a *V*_{g}-*B* relation enables us to deduce the energy width ∆*E*_{N} of the extended states of the *N*th Landau level. Figure 1(c) and (d) show temperature (*T*) dependences of ∆*E*_{0} and ∆*E*_{1}. These show that ∆*E*_{0} and ∆*E*_{1} have similar magnitudes and are proportional to *T*^{η} with *η* = 0.30 - 0.31 for ∆*E*_{0} and *η* = 0.32 - 0.35 for ∆*E*_{1}. These values are comparable to the critical components previously deduced from the plateau-plateau transition in quantum Hall regimes. Moreover, we deduced the energy widths independently using activation gap measurements. The energy width for *N* = 1 Landau level show a good agreement between two different types of measurement methods, while the energy width for the *N* = 0 Landau level obtained from activation gap measurement is larger by 30 meV than that deduced from the transport energy-spectroscopy.

Using these two different measurement techniques systematically, we demonstrate that our device include random disorder rather than typical hopping disorder such as ripples.

This work was partly supported by KAKENHI.

- [1] K. Takase, S. Tanabe, S. Sasaki, H. Hibino, and K. Muraki, Phys. Rev. B
**86**, 165435 (2012). - [2] K. Takase, H. Hibino, and K. Muraki, Phys. Rev. B
**92**, 125407 (2015).

Fig. 1. (a) Longitudinal resistance as a function of gate voltage (V_{g}) and magnetic field (B). (b) Energy diagram of graphene with interface states. Temperature dependence of ∆E for the_{N} N = 0 Landau level (c) and N = 1 Landau level (d). |