Graphene is a semi-metal, or a zero-gap semiconductor because its conducting and valence π electron bands touch each other only at two isolated points in its two-dimensional (2D) Brillouin zone [

12]. The conical shape of these bands in the vicinity of these points gives rise to an approximately linear density of states,

, where

*g*_{d} = 4 is the spin and the band valley degeneracy factor, and

*v*_{F} ≈

*c*/300 is the Fermi speed of graphene, with

*c* being the speed of light in vacuum [

12]. In the intrinsic, or undoped graphene, the Fermi energy level sits precisely at the neutrality point, ɛ

_{F} = 0, also called the Dirac point. Therefore, the electrical conductivity of graphene is easily controlled, e.g., by applying a gate voltage

*V*_{A} that will cause doping of graphene’s π bands with electrons or holes (depending on the sign of

*V*_{A}), which can attain the number density per unit area,

*n*, with a typical range of

*n* ∼ 10

^{11}–10

^{13} cm

^{−2}[

12]. In a doped graphene, Fermi level moves to

, where sgn(

*n*) = ± 1 for electron (hole) doping. At a finite temperature

*T*, one can express the charge carrier density in a doped graphene in terms of its chemical potential μ as [

19]

where
with *k*_{B} being the Boltzmann constant. We shall use in our calculations a full, non-linear expression for the π electron band density, ρ(ɛ), given in Eq. 14 of Ref. [12]. However, for the sake of transparency, the theoretical model for graphene will be outlined below within the linear density approximation, ρ(ɛ) ≈ ρ_{L} (ɛ). We note that this approximation is accurate enough for low to moderate doping levels, such that, e.g.,
, and it only incurs a relative error of up to a few percent when
.

At this point, it is convenient to define the potential

*V*_{Q} = −μ/

*e*, where

*e* > 0 is the proton charge, which is associated with the quantum-mechanical effects of graphene’s band structure [

20], and relate it to the induced charge density per unit area on doped graphene, σ = −

*en*, via the Eq.

1,

where dilog is the standard dilogarithm function [

21]. One can finally use the definition of differential capacitance per unit area,

*C*_{Q} = dσ/d

*V*_{Q}, to obtain from Eq.

2 the quantum capacitance of a single layer of graphene as [

14]

where we have defined the characteristic length scale for graphene,

with the value of λ_{Q} ≈ 18 nm at RT. Note from Eq. 3 that graphene’s quantum capacitance grows practically linearly with *V*_{Q} when this potential exceeds the thermal potential, *V*_{th} = 1/(*e* β), having the value of ≈26 mV at RT.

We further assume that an upper surface of graphene is exposed to a thick layer of a symmetric

*z*:

*z* electrolyte containing the bulk number density per unit volume,

*N*, of dissolved salt ions. Taking advantage of planar symmetry, we place an

*x* axis perpendicular to graphene and pointing into the electrolyte. The theory developed by Borukhov et al. [

22,

23] to model finite ion size uses the mPB equation for the electrostatic potential

*V* (

*x*) in the electrolyte at a distance

*x* from graphene, given by

where

*z*(=1) is the valency of ions, ε is relative dielectric constant of water (≈80, assumed to be constant throughout the electrolyte), and γ = 2

*a*^{3}*N* is the packing parameter of the solvated ions, which are assumed to have same effective size, equal to

*a*[

22,

23]. We note that the standard PB model is recovered from Eq.

5 in the limit γ → 0 [

13]. By assuming the boundary condition

*V*(

*x*) = 0 (and hence d

*V*/d

*x* = 0) at

*x* → ∞, deep into the electrolyte bulk, Eq.

5 can be integrated once giving a relation between the electric field and the potential at a distance

*x* from graphene. Assuming that graphene is placed at

*x* = 0, one can use the boundary condition at the distance

*d* of closest approach for ions in the electrolyte to graphene,

to establish a connection between the induced charge density on graphene, σ, and the potential drop,

*V*_{D} =

*V*(

*d*), across the EDL as

The total potential,

*V*_{A}, applied between the reference electrode in the electrolyte and graphene can be written as

where *V*_{pzc} = (*W*_{gr} − *W*_{ref})/*e* is the potential of zero charge [13] that stems from difference between the work functions of graphene and the reference electrode, *W*_{gr} and *W*_{ref}, respectively, and *V*_{cl} = 4πhσ/ε′ is the potential drop across a charge-free region between the compact layer of the electrolyte ions condensed on the graphene surface, having the thickness *h* on the order of the distance of closest approach *d*[23, 24], and with ε′ < ε taking into account a reduction of the dielectric constant of water close to a charged wall [25]. In our calculations, we shall neglect these two contributions to the applied potential in Eq. 8 because *V*_{pzc} merely shifts the zero of that potential, while a proper modeling of *V*_{cl} involves large uncertainty [23, 24]. However, usually the effects of *V*_{cl} can be considered either small [23] or incorporated in the mPB model via saturation of the ion density at the electrolyte–graphene interface for high potential values [22]. Consequently, *V*_{D} and *V*_{Q} represent the two main contributions in Eq. 8, with *V*_{D} being the surface potential of graphene that shifts its Dirac point, and *V*_{Q} being responsible for controlling the doping of graphene by changing its chemical potential. Finally, we note that all results of our calculations will be symmetrical relative to the change in sign of the applied potential because of our assumption that the effective sizes of the positive and negative ions are equal [22, 23], but this constraint can be lifted by a relatively simple amendment to the mPB model [24].

Using the relation

*V*_{A} =

*V*_{Q} +

*V*_{D}, we obtain the total differential capacitance of the electrolytically top-gated graphene as

where

*C*_{Q}(

*V*_{Q}) is given in Eq.

3, and

*C*_{D}(

*V*_{D}) = dσ/d

*V*_{D} is the differential capacitance per unit area of the EDL, which can be obtained from Eq.

7as [

23,

24],

with

being the Debye length of the EDL [

13]. Note that, in the limit of a very low potential

*V*_{D}, and hence for low density of ions at the graphene–EDL interface, one can set γ → 0 in Eq.

10 to recover an expression for the EDL capacitance in the standard PB model [

13],

We further note that, while Eq. 11 implies an unbounded growth of the EDL capacitance with *V*_{D} in the PB model, Eq. 10 suggests a non-monotonous behavior that will eventually give rise to a saturation of the total gate capacitance at high applied voltages.