The paper you referenced states that the potential ##U## is found by solving Poisson's equation:
"The variation of carrier concentration has a marked influence on the band structure as derived from a comparison of the experimental results with the tight-binding calculation (Fig. 4). The Coulomb potential difference U displays a sign change at the electron concentration where the gap closes. It is expected that U increases with an increase of the charge difference in either graphene layer induced by the fields at the respective interfaces. We have calculated the potential of each graphene layer from Poisson’s equation based on the Schottky barrier height of 0.4 eV (15) assuming infinitely thick graphene multilayers, and find that for the as-prepared sample the potential difference between the first and second layers shows reasonable agreement with the Coulomb potential difference U estimated from the size of the gap evaluated in the tight binding model." (emphasis added)
But lacking the details of the charge distribution and boundary conditions assumed by the authors, it's anyone's guess how they arrive at their Fig. 4B:
View attachment 354580
Nevertheless, you can at least approximate their result by modeling the two layers of graphite as a simple parallel-plate capacitor, for which the voltage ##V## between the plates is given by the well-known formula ##V=\frac{Qd}{\varepsilon_{0}A}+V_{0}\,##. Here ##Q## is the charge-per-unit-cell (in ##\text{coulombs}##, found from the horizontal axis of Fig. 4B), ##\text{d}## is the separation distance (in ##\text{m}##) between the graphite layers, ##\varepsilon_{0}## is the permittivity of vacuum (in ##\text{farads per m}##), ##A## is the unit-cell area (in ##\text{m}^2##) and ##V_0## is a constant background voltage. Then ##U=q_e V##, where ##q_e## is the coulomb-charge of the electron. (Note that this simplified equation predicts a linear relation between ##U## and ##Q## and thus it fails to model the "roll off" in the upper right corner of Fig. 4B.)