Fig. 3. Dielectric breakdown and Pauli blocking operation.

a Maximum Fermi energy, noted $E_{\mathrm{F}}^{\max }$, expected at the graphene electrodes of a graphene modulator with a dielectric’s relative permittivity ϵ_r and dielectric strength E_BD. All points lying inside the blue-colored region represent a dielectric allowing for Pauli blocking operation ($E_{\mathrm{F}}^{\max }>0.5$ eV, refer to Supplementary Note 1). The red-colored region indicates otherwise ($E_{\mathrm{F}}^{\max }<0.5$ eV). The white band represents the Pauli blocking boundary condition, defined as $E_{\mathrm{F}}^{\max }=0.5$ eV. The expected $E_{\mathrm{F}}^{\max }$ for HfO₂ and hBN are represented by the red and green squares, respectively, taking the values of E_BD and ϵ_r from literature^28,35–37 (marked with dots) and our dielectric characterization (marked with stars, see Supplementary Notes 5 and 10). The black star represents the $E_{\mathrm{F}}^{\max }=0.57$ eV expected for the hBN–HfO₂–hBN modulator in Fig. 1e (see Supplementary Note 10). b, c Normalized transmission as a function of E_F and V_BT for modulators with hBN (b) and hBN–HfO₂–hBN (c) dielectric. The data points are measurements and the solid curves simulations (see Supplementary Notes 1–3 and 10). The vertical dashed lines indicate the $E_{\mathrm{F}}^{\max }$ achieved at the dielectric breakdown. The orange-shaded regions show the full transparency range, i.e., Pauli blocking. The top V_BT axis in panel b is for the 42 μm-long device only (see Supplementary Note 7 for the other hBN devices). The graphene Dirac cones in panel b show the absorption and Pauli blocking processes at low and high Fermi energies, respectively.