Exfoliation had to lead to a contacted sample
Novoselov, Geim and colleagues reported electric-field effects in atomically thin carbon films in 2004. Mechanical exfoliation supplied flakes; locating thin regions and making contacts allowed their electronic behavior to be measured.
The 2010 award recognizes experimental work on graphene. The achievement includes turning a fragile-looking sheet into a reproducible material system, not merely drawing or naming a two-dimensional lattice.
A gate changed the conducting carriers
A gate voltage changes the charge density of the sheet through electrostatic coupling. Transport can pass between electron-dominated and hole-dominated behavior. Contacts, substrate disorder and residual charge complicate the response near charge neutrality.
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Two-dimensional bands invite further tests
The low-energy band structure of monolayer graphene motivates a Dirac description, but a field-effect curve alone does not establish every consequence of it. Magnetic oscillations and quantum Hall measurements provide additional tests of the electronic spectrum.
The linked models show band concepts
The periodic-band and Chern-insulator Labs provide controlled examples of band motion and integer topology. Their Hamiltonians describe their own systems. They should not be read as measurements of exfoliated graphene or as a prediction of a commercial graphene transistor.