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96 Chapter 2

groundbreaking research on the exceptional property of the material. They produced their
graphene with pencil markings and scotch tape. By repeatedly sticking fresh bits of tape against
a pencil mark, the graphite layer became thinner and thinner until a single sheet of carbon
remained. The “Gold Rush” of graphene began and continued to this day. More than 8000
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papers (~22 papers/day) about graphene were published in 2012 alone. The unique properties
of graphene have motivated intense work among physicists and engineers who have seen in this
material new opportunities to improve low noise amplifiers, frequency multipliers, mixers,
resonators, biology sensors, ultra-thin films transparency in microwave and optical spectrum,
low-power switches, displays, touchscreens, light-emitting diodes, solar cells, supercapacitor
energy storage, water filtration and desalination, corrosion and ice formation prevention, and
many more just undiscovered prospects. Graphene has been identified as one of the most
promising materials for post-silicon electronics.

o
The graphene melting point is around 4900 K, its layer is about 100 times stronger than steel,
extremely stretchable, chemically stable, incredibly thin (0.34 nm) and thus supernaturally
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lightweight (the sheet of 1 m weights 0.77 mg = 0.00000077 kg (!)), optically opaque absorbing
only 2-3% of incident light, exceptionally flexible and stretchable (up to 20% of initial length),
and 65% more electroconductive than copper. The thermal conductivity of graphene layer is
around 3080 - 5300 [W/(m∙K)] while that of diamond is around 1000 [W/(m∙K)]. It means that
it can be used as a bridge with low thermal resistance between chips and the thermal sink.
Besides, it is highly hydrophobic tending to repel water from its surface. Additionally, theory
predicts a whole spectrum of beneficial magnetic phenomena in graphene.

Explanation on the unique physics of graphene is based deeply on quantum phenomena and far
beyond the scope of this book. That is why we focus below on the graphene as an engineering
material, and some technologically advanced applications are referring the reader for details to
the specialized literature [3-5]. We believe that our reader using these data may develop new
graphene devices and explore new application areas for them.

Current prices for graphene production is relatively high. For example, 75×75 mm single layer
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of graphene on copper foil costs around $1000 . Nevertheless, graphene fabrication has only
been getting better since the material was first isolated. Almost every day the publications
promise to make graphene manufacturing 100 and even 1000 times cheaper.

2.9.2 Graphene as a Conductor

Recall that we introduced through the expression (2.28) the macroscopic conductivity
parameter as the coefficient relating the volume electric current density with E-field that

initiates this current and vice versa. If so, such conductivity description is relevant for common
3D materials. Meanwhile, a graphene sheet is 2D one-atom structure of 0.34 nm thick that
makes this 3D quantity transcendental. Furthermore, as we have mentioned before, the
conductivity is really provided by “extra thin” layers of pi-electrons clouds on both sides of the
graphene lattice. The best way to avoid all these troubles is to include the imaginary graphene
thicknesses into the conductivity definition introducing the surface conductivity = ∙ =

1/ [S] (see (2.40)). The quantum physic approach expresses such conductivity without any
□
mention of thickness.

24 Rajni Garg, Naba K. Dutta, Namita Roy Choudhury, “Work Function Engineering of Graphene,”
Nanomaterials 2014, v. 4.
25 http://shop.grapheneplatform.com/custom/