Fig. 1

Layer polarization of bilayer graphene at zero magnetic field. a Measurement schematic showing geometric gate capacitances c _t and c _b and interlayer capacitance c ₀. Capacitance is measured using a cryogenic bridge circuit by comparison with a standard capacitor C _std, measured to be 404 ± 20 fF (see “Methods”). b Device image. Top gate (TG), back gate (BG), and contacts to bilayer graphene (G) are shown. Scale bar is 10 μm; device area is approximately 87 μm². c C _S measured at B = 0 and T = 1.6 K as a function of n ₀/c = v _t + v _b and p ₀/c = v _t − v _b. A p ₀-dependent band gap is visible as the dark region near n ₀ = 0. d Line traces taken at different values of p ₀, corresponding to dashed lines in c. Band edge van Hove singularities²⁸ and electron-hole asymmetry²⁷ are both evident. e C _A measured under the same conditions. A common, constant background has been subtracted to account for fixed parasitic capacitances. f Line traces at different values of p ₀ corresponding to dashed lines in e. g Integrated change in polarization, \(\frac{{{c_0}}}{c}{\int} {{C_A}{\kern 1pt} d\left( {\frac{{{n_0}}}{c}} \right) = \Delta p} \), with the constant of integration fixed to be zero at high \(\left| {{n_0}} \right|\). In accordance with single-particle band structure²⁸, wavefunctions are layer unpolarized for p ₀ = 0, while for large \(\left| {{p_0}} \right|\) the polarization peaks at n ₀ = 0, where band wavefunctions are strongly layer polarized