CHAPTER 1. INTRODUCTION
1.5 Graphene
Graphene is a two-dimensional flat monolayer of carbon atoms arranged in a
honeycomb structure and is actually a different name for the basal plane of graphite. It was
argued that strictly 2D crystals could not exist due to large thermal fluctuations that would
disrupt a 2D structure until graphene and single-layer boron nitride were isolated and
placed on non-crystalline substrate like SiO2, on top of a liquid layer or prepared as a
Of numerous special properties of graphene mentioned in the beginning of this section,
DOS around the K-point for bulk of graphene and of its edges deserves special attention when considering electrochemical behaviour of graphene. Graphene is classified as a zero-
gap semiconductor25 due to the fact that its valence and conduction bands touch each other
at one common point named the Dirac point. It is unique in having (to a good approximation) linear dependency of DOS,* denoted ρ, on energy around the K-point:25,39
2 F 2 1 ( ) ( ) v (1.1)
where vF is the Fermi velocity and energy ε is counted from the Dirac point. Graphene
owes its special band structure to the fact that it is made of identical atoms. Its nearest
analogue is boron nitride which also has a flat honeycomb arrangement of the atoms but it
is an insulator with the gap of 6 eV.39
Figure 1.2. Schematically shown are DOS of pristine undoped graphene (dashed black) and “dull” DOS around the Dirac point of graphene on SiO2 (solid blue). Adapted with modifications from ref39
In all electrochemical measurements done on graphene, a solid substrate was used to
support extremely fragile graphene film. It is difficult to imagine electrochemistry on a
suspended graphene membrane. As mentioned earlier in this chapter, the substrate was
shown to have an effect on redox properties of graphene. Frequently used oxidized silicon
wafers, are covered with a layer of amorphous SiO2 that has on its surface regions of
positive and negative charges (charged impurities), which makes the electrical potential to
vary over the surface. The concentration of charge carriers in graphene is very sensitive to
the external electric field since the conduction and valence bands just tough each other.
Thus, variable surface potential of SiO2 induces local variations of the doping level in
graphene that overall smears the Dirac point, effectively making it “duller” as
schematically shown in Figure 1.2.39 Whether this effect is critical for the electrochemistry
of graphene electrodes in general or limited to only some of redox reactions is yet to be
established.
Graphene’s low DOS around the intrinsic Fermi level leads to the appearance of
quantum capacitance in this material. This term was introduced by S. Luryi when graphene was still a theoretical model and figured as a two-dimensional electron gas in that
early work.94,95. Quantum capacitance originates from the Pauli’s exclusion principle
applied to a quantum model of 2D electron gas. Specifically, it states that filling the
quantum well with electrons requires extra energy.94 Also, capacitance of the systems
composed of low-DOS materials cannot be explained entirely by considering the geometry
and potential difference across the plates; the electronic structure needs to be taken into
account as well.
In the case of graphene, the effect of quantum capacitance can be schematically
explained as follows.96 Consider an electrolyte top-gated graphene electrode (Figure 1.3).
Its Fermi level sits exactly at the Dirac point when the electrode is held at the potential of
zero charge (PZC)* and is EPZC volts below the level of a reference electrode. When it is
* this is idealized presentation and doping of graphene due to contact with solution and/or substrate is
negatively biased relative to the PZC (potential difference is E1), the electrode takes
electrons that fill the conduction band from the bottom. The upper level of electron energy
counted from the bottom of the conduction band is called band filling potential of graphene, φbf.97 Since the electrode acquired negative charge, it changed all the levels of
electron energy viz the band structure “went up” on the energy scale. This shift of the band structure can easily be seen in the figure as a displacement of the Dirac point. The
magnitude of this displacement equals exactly the potential drop across the double layer,
ΔEEDL, since it has pure electrostatic origin. When graphene is positively biased relative to
the PZC, the whole band structure shifts down the energy scale also by ΔEEDL and
additionally holes fill the valence band. The potential difference between the reference
electrode and the graphene (the difference between the Fermi levels of the two electrodes)
is E2 in this case. It differs from pure electrostatic potential difference by φbf.
Figure 1.3. Schematic presentation of quantum capacitance in electrolyte top-gated graphene. Adapted from ref96 with modifications. RE = reference electrode and Gr = graphene.
In the case of metal electrodes, DOS is so large that filling/empting energy levels with
difference. If q is the charge (per unit area) that the graphene electrode acquired then quantum capacitance is defined by eq (1.2):96,97
bf Q q C φ (1.2)
From this equation, it is clear that, because for metals Δφbf ~ 0 for any finite Δq, CQ is
large and its contribution to the total capacitance is effectively zero.
Another feature of graphene (and graphite) important for electrochemical studies
(especially in view of the earlier introduced theory overemphasizing the significance of
step edge in electrochemical activity) is the edge state. It was predicted theoretically98 and
corroborated experimentally with STS/STM measurements99 that DOS is significantly
enhanced at the Fermi level along the zigzag edges and to some extent mixed (more real)