The role of defects and dimensionality in influencing the charge, capacitance, and energy storage of graphene and 2D materials

2.1 Double layer capacitance

The placement of a 2D material (on an electrode, for measurement) into an ambient would be associated with a charge separation due to the respective differences in the electrochemical potential and yields a net capacitance, from two contributions, i.e. a Helmholtz capacitance and a diffusion capacitance. For example, when the electrode is positive (negative), it would be surrounded by corresponding negative (positive) charge. The adjacent oppositely charged layers constitute a double-layer [11], [12], [13], [14] capacitance. The resultant Helmholtz capacitance, C_H (per unit area), for the 2D material electrode, with charge stored in a layer of thickness (d) is given, in an elementary form, by the following:

In Eq. (1), ε=ε_oε_r, with ε_o as the permittivity of free space and ε_r as the relative dielectric permittivity of the ambient. While d has conventionally been taken as the average separation distance between the positive and negative charges, a more precise equivalence would be the Thomas-Fermi screening length, λT-F(=2nεe2/EF), which is related to the span over which the electrical carriers in the 2D material (of density, n, with e as elementary electron charge, and Fermi energy, E_F) exert their influence into the ambient [15].

2.2 Diffusion capacitance

Additionally, there would be a spread of the counteracting carriers/ions from the ambient away from the material, and the ratio of the resultant charge distribution and the potential variation yield the diffusion capacitance (C_D). The combination of C_D and C_H, in series, is generally considered as the net double-layer capacitance (C_dl). C_D is of the following form [16]:

The above relation (with I^o as the ambient ion concentration and applied voltage/potential (ϕ) and with z as the magnitude of ion charge, e.g. +1 for H⁺/Na⁺ and −1 for OH⁻/Cl⁻, k_B is the Boltzmann constant and T is the temperature) is obtained through solving the Poisson-Boltzmann equation, with ambient carrier concentration varying as exp (−eϕ˜kBT). Concomitantly, Debye length (L_D) [16], [17] is considered to be a typical measure of the diffuse layer thickness constituting C_D and is given by the following:

However, the cosh () term in Eq. (2) seems to indicate an increase of the capacitance without limit as the voltage (/ϕ) is increased. Such a conclusion is contrary to typical experimental observation where the saturation of the measured capacitance is observed with increasing voltage. Generally, at sufficiently large applied potentials, oppositely charged ions would indeed adhere strongly to the 2D material, with an average separation distance subject to limitations arising from ionic radius as well as the influence of the ambient (e.g. the solvation of the ions in aqueous systems). As I^o or ϕ is increased, at a given temperature, C_D>C_H, and the diffusion layer is increasingly irrelevant for the measured capacitance and can be discriminated. It may also be noted, for example, that at smaller ϕ, C_dl→C_D; at ϕ (of the order of 3k_BT), C_H and C_D are comparable; and at a larger ϕ(greater than 10 k_BT), C_dl→C_H. There are yet issues in modeling the precise magnitudes of C_D and C_H. It is unclear, for example, whether the bulk ambient dielectric permittivity [18] would be appropriate for distances close to the 2D material-ambient interface. The large electric field at the surface (due to the potential drop over a size scale of an ion) implies the necessity for considering orientational effects, e.g. of the water molecules from the ambient, in addition to enhanced polarization [19]. Consequently, ε_r may be considerably reduced, as much as an order of magnitude from the bulk value, e.g. in the case of ambient moisture/water, from ~78 to as low as 4.7 for H⁺/OH⁻ ion-2D material distances of the order of 0.1 nm (note that the radii of H⁺ and OH⁻ ions is ~0.09 nm and 0.155 nm, respectively). Such reduction in ε_r correspondingly reduces C_H and complicates understanding effects of ambient water on 2D materials [20]. Further experimental work is necessary to yield relevant insights into such issues.

2.3 Quantum capacitance

A characteristic particular to low dimensional materials, such as 2D graphene or one-dimensional (1D) nanotubes [5], [21], is the finite DOS. Consequently, there is a relatively larger increase (decrease) of Fermi energy (E_F) when electronic charge of magnitude dQ (=e·dn) is added (removed) due to an applied voltage change (dV) [22]. An effective quantum capacitance (C_Q) in series with the C_dl could be defined, considering DOS at the E_F, as follows:

A low C_Q generally indicates lack of available states that can be occupied, e.g. during charging and influences device characteristics. Broadly, a net applied voltage (ΔV) causes a differential change in E_F and would be partitioned between the 2D material (as ΔV_Q) and the surrounding ambient/environment (ΔV_E). While ΔV_Q would be associated with C_Q, ΔV_E is related to additional classical capacitances due to the internal and external environment of the 2D material. While C_dlis a manifestation of the latter, an aspect of internal charge re-distribution would be through a space-charge capacitance (C_SC) [23], [24], [25].

2.4 Space charge capacitance

An additional capacitance in series with C_dland the C_Q, particular to layered 2D nanostructures such as FLGs, is C_SC, which arises due to the screening of the ambient charge into the inner layers by the outer layers, see Figure 2. With a carrier concentration (n) of the order of 5×10²⁰/cm³, the screening distance would be ~0.34 nm – the thickness of a graphene sheet [3], [28]. The implication is that an atomic layer sheet would not completely screen the electric fields and a single-layer 2D material could not constitute a perfect electrode. The consequent internal charge diffusion and re-arrangement in the 2D material (with ε=ε_oε_r and with ε_r as the relative permittivity of the 2D material) implies a C_sc [29] as follows:

Figure 2:

While the constituent charge in the individual layers of a 2D material may be considered discrete, the carrier concentration from the outermost layer into the current collector may be modeled through a continuous exponential decay. A concomitant length scale for the internal charge storage would be an equivalent Thomas-Fermi screening length [26] (λT-F). Additionally, immobile surface charges (due to defects) may also contribute to the space charge. (Figure adapted from Reference [27] and reproduced with permission from the American Chemical Society, 2015.)

While such a form seems similar to that of Eq. (2), n_2D,1 now refers to the carrier concentration in the first/outermost layer of the 2D material, immediately adjacent to the ambient, the charge of which is being screened into the other interior layers of the 2D material. Alternately, a single-layered 2D material (e.g. SLG) would not have a spread of charge into the interior and no C_SC. This would help in the further delineation of the constituents of the measured capacitance. Generally, the consideration of C_SC would also be very relevant for multiple-layered 2D materials (such as FLGs, MoS₂, WS₂, Mo₂P, etc.).

2.5 Chemical capacitance

The net electrochemical potential (the Fermi energy: E_F) constituted the electrostatic potential (due to carrier density change at a point) and the chemical potential (due to carrier density being distributed over a representative volume element), yielding electrostatic capacitance and chemical capacitances, respectively. While the former has been previously discussed (e.g. through C_H,C_D, C_SC, etc.), the latter is manifested, for example, in a photo-electrochemical cell, as used for photo-electrochemical applications [30], [31], [32]. Consider, for example, a macroscopic capacitor constituting nanoscopic units arranged in a mesoscale architecture, e.g. TiO₂ nanoparticles on the electrode of a dye-sensitized solar cell (DSSC) [33]. The potential difference between the electrodes of the capacitor would be mostly concentrated on the nanoparticles and the interfaces, due to their large number. A chemical capacitance (C_chem) specific to such mesoscopic architecture may been defined [34] through the following:

The consideration of the nanostructured entities is done through the change of the number of charge states: N_i (for the i^th nanoparticle/grain) corresponding to both the free electrons (of concentration, n_c) and localized/trapped electrons (of concentration, n_T for a given change in the chemical potential: μ_i). From the thermodynamic expression [35] μ_i=μ_i^o+k_BT ln (N_i), we may derive, e.g. C_chem=e²N_i/k_BT.

Thus far, the individual contributions to a measured capacitance (C_meas) have all been manifested through series contributions of the form (see Figure 3):

Figure 3:

A possible impedance model consisting of additional resistance and capacitance contributions in 2D material that constituted nanostructured electrodes. In non-metallic materials, space charge capacitance – C_sc (see Section 2.3), quantum capacitance – C_Q (see Section 2.4), in addition to a Faradaic/surface area-dependent pseudocapacitance or chemical capacitance – C_p(see Section 2.5), as well as concomitant leakage/shunt resistances – R_sh, may be present.

It is obvious, from the relation above, that the lowest capacitance would dominate C_meas, e.g. through a low C_Q for 2D materials of the order of ~2 μF/cm² [36], [37], [38], [39] due to the very low DOS. Alternately, a very high capacitance – of the order of 500 μF/cm² – is typical of C_H, from Eq. (1). Moreover, 2D material based nanostructures have been postulated to have a theoretical surface area per mass of the order of, e.g. ~2600 m²/g for graphene [40], [41]. Consequently, the theoretical maximum possible capacitance density may be thought to be of the order of 13,000 F/g (from the product of 500 μF/cm² and 2600 m²/g). However, the reported/measured capacitances are orders of magnitude lower [42], at ~100 F/g [43], [44], [45], [46], [47]. It can now be understood that low measured capacitances arise from the (a) series addition of various classical and quantum capacitances, as described through Eq. (6), as well as (b) incomplete/inadequate utilization of surface area [48].

Moreover, we specifically address the influence of the real surface area contributing to the charge storage. C_meas would be expected to be directly proportional to such an area. The utility of edge planes, which do not directly contribute to the projected area but which have been hypothesized as contributing to the charge density [49], [50], [51], [52], could thus be verified.

4.1 The extent of the surface area in nanostructured materials and electrodes

A major issue in quantifying and correlating the capacitance to theoretical estimates is the accurate knowledge of the effective area (A_eff) of charge transfer, such that the relevant unit capacitance (i.e. capacitance per unit area: C_Area) can be rigorously determined. While conventionally, the projected area of a 2D material that constituted a substrate is considered, more accurate and precise determination of the area is generally warranted to consider the (a) different charge storage capacities of the constituent moieties (e.g. to determine the extent to which Mo and S in MoS₂ store charge), (b) multi-layered 2D materials and constituent edges, (c) defects and adsorbed moieties, which may/may not participate in charge transfer, etc. Such characterization is crucial in that the capacitance density (capacitance per unit mass: C_Mass) is often estimated by multiplying C_Area by the areal density (in units of m²/g) – determined through gas adsorption analysis. It is then seen that there is much disparity and consequent confusion in reported values of C_Mass in literature, as (i) gas adsorption analysis overestimates the available area, gas molecules have greater access to nanostructure porosity compared to electrolyte molecules/ions, and (ii) the projected (and not the actual) area is often used for evaluating C_Area. It is then required to develop accurate metrics and methodologies for reporting C_Area as well as C_Mass, with an aim to measure and crosscheck the real A_eff of 2D material structured electrodes. The area may be estimated, e.g. through both the experimentally measured maximum/net current considering the major possible electrochemical processes [16], [17], as well as through a complementary methodology, employing integrated currents (i.e. charge) to verify the A_eff value obtained above. Here, as an initial voltage step, the total charge passed to the solution (Q^tot) could be defined through the sum of the charges relevant to (i) diffusion related processes (Q_diff), (ii) adsorbed species (Q_ads), and (iii) double layer (Q_dl) [17]. As charge transfer to the adsorbed species (Q_ads) and the double layer (Q_dl) occur on a relatively shorter time scale than charge transfer arising from diffusion processes, Q_diff can be isolated from the effects related to Q_dl+Q_ads allowing for the estimation of A_eff.

The use of nanostructured materials in electrodes would also pose problems related to the extent to which the electrolyte is fully in contact with the constituent area, for charge transfer interactions to occur. Such issues may be discussed with respect as to whether the electrolyte wets the electrode and is a function of the respective surface energies as well as physical characteristics such as the roughness of the electrode, etc. Classical monographs [83] describe such interactions fundamentally by considering a wetting/spreading parameter (Sp) of the form Sp=γ_s-a–(γ_s-e+γ_e-a), where the subscripts refer to the relative surface energies of the solid electrode-air, solid electrode-electrolyte, and the electrolyte-air interfaces, respectively. Generally, when Sp>0, the electrode prefers to be wetted by the electrolyte and good contact would be ensured. It was postulated that idealized interactions between the individual constituents could make it possible to relate Sp to the individual polarizabilities (surface energies being related to the square of the polarizability [62]). Consequently, an electrolyte would spread on/wet completely the electrode, if it was less polarizable than the solid electrode. Concomitantly, it can be adduced that enhancing the polarization of the electrode, e.g. as in metallic materials or through the edge planes in graphite, or defects would favor greater contact between the electrode and electrolyte. A prototypical example of roughness at the nano-, meso-, and macro-scale is practically manifested through porous carbon/AC-based electrodes in ECs.

The relative dielectric constants (ε) of the electrolyte (ε_e) and the material constituting the electrode (ε_el) should also be considered. For aqueous electrolytes, one should also consider the relative hydrophobic and hydrophilic character of the surfaces, e.g. the latter may be enhanced through the presence of polar moieties and charged defect states. On the other hand, hydrophobic surfaces may be stochastically wetted/de-wetted by water for pore sizes of the order of 0.7–0.8 nm [84]. A ratio of the actual area to the apparent projected area has been used to describe the wetting characteristics and may also be extrapolated to estimate the enhanced charge density of a rough surface. Alternatively, another quantitative methodology involves the use of the current-potential curves obtained from surface adsorption based pseudocapacitance [85], e.g. the ratio of the theoretical maximum capacitance to the measured capacitance would yield a measure of the roughness to within ±40–50% [85]. A roughness factor has also been used in estimating the capacitance of nanostructured electrodes, e.g. in dye-sensitized solar cells (DSSCs). Here it was assumed that the Helmholtz capacitance (C_H) would be similar for any two surfaces and the roughness factor is used to multiply the capacitance of a smooth surface to obtain the capacitance of the rough surface [86]. Additionally, corrugations and the local curvature of the surface could simulate pore-like behavior and influence the observed characteristics. While the complexity of the problem and the general irreproducibility across samples have resulted in ignoring such issues, roughness yet remains a very important issue in understanding the effective area of a nanostructured material constituting electrode as well as for correlating the theoretical predictions to experiments.

4.2 Issues due to energy discretization in nanostructures

A major issue in harnessing charge from nanostructures is ensuring compatibility of the associated energy scales between the nanoscale – where the charge is stored – and the macroscale – to which the charge is transferred. If there is an incompatibility, e.g. considering the difference in the DOS, where a discrete distribution is obtained for the nanoscale and a continuous distribution for the macroscale, an energy barrier [29] would exist with the technological implication of different times for charging and discharging the energy storage device. Only in those cases where the equivalent width of the barrier region is sufficiently small may carriers tunnel through yielding a quasi-Ohmic behavior and equivalent times for charge/discharge. Such behavior has been observed, and typically for nanostructure lengths <1 μm, electrical transport does seem to be limited by the Schottky barriers at the metal-semiconductor interface. However, for greater lengths and in lower dimensionality, there would be a stronger influence of the defects/impurities (as outlined in Section 3), which may induce a strong resonant backscattering and reduce carrier mobility [87]. Such defects in nanomaterials would also contribute significantly to a combined capacitive and battery-like behavior, as has been found in large specific area aerogels [88] and carbon nanotubes [40], [89], [90]. An example of the practical necessity for the consideration of quantum effects is the manifestation of an additional quantum capacitance (C_Q) – in series with the electrostatic capacitance, which is directly proportional to the DOS of the pertinent nanostructures – as discussed in detail in Section 2.3. The series addition implies that the C_Q needs to be larger than the nominal electrostatic contribution for its neglect, which would typically not be possible, as nanostructures intrinsically possess a smaller number of states with a smaller C_Q.

In addition to the scientific issues related to the adequate utilization and parameterization of the surface area as well as the finite DOS due to nanostructuring discussed in the previous two sections, there are a host of more practical problems related to the use of nanostructured electrodes. While a comprehensive discussion of such problems, which may include low packing densities (which contributes to low volumetric energy density in distinction to gravimetric energy density), undesirable parasitic reactions with the electrolyte (stemming from the large surface area to volume ratio of the constituents of the electrodes), and high manufacturing costs are outside the scope of the present review, the reader is encouraged to consult a paper [91] related to charge transfer and storage in nanostructures, which considers such aspects in more detail. For example, while 1D nanostructures have a higher electronic DOS at lower energies compared to bulk materials, the net amount of electrons could be smaller. As macroscopic properties such as electrical conductivity would be proportional to the net amount, nanostructured electrodes could have inferior electrical characteristics. Alternately, if 1D nanostructures could be packed tightly so as to occupy the same volume as the bulk, there would then be the possibility of higher electrical conduction efficiency (due to lower scattering in 1D channels), but such aspects are difficult to implement in a viable manufacturing process. The disadvantages accrued from non-close packing would be manifested in the accumulation of undesirable ambient impurities. Indeed, such extrinsic aspects may circumscribe the extent of possible usage of the nanomaterials even more than the possible advantages that may accrue from the lower dimensionality.