388
13 Two-Dimensional Semiconductors
Fig. 13.7 Cyclotron mass
in graphene as a function of
the sheet electron
concentration n S (negative
values relate to hole
concentration, E F < E D ).
Adapted from [1249]
N (E F ) = 4
π k
2
F
(2π/L) 2 = A
4π E
2
F
h 2 v
2
F
,
(13.6)
where A is the system area. Therefore we have (at low temperature) with (13.5)
n S =
4π
h 2
E
2
F
v
2
F
∝ m
2
c ,
(13.7)
as determined experimentally. Therefore the behavior of the SdH oscillations confirms the linear
dispersion relation. The experimental value for the velocity is v F ≈ 10
6 m/s. From (13.7) the density
of states (per area and energy) around the Dirac point increases linearly with energy,
D(E) =
8π
h 2 v
2
F
E .
(13.8)
The carrier density in a graphene sheet can be controlled via the field effect. The graphene is positioned
on an insulator/semiconductor structure, typically SiO 2 /Si (cf. Sect. 21.3). The carrier density is then
related to the applied (gate) voltage V g via (21.93) and (21.95), i.e.
n S =
i V g
e d
,
(13.9)
where d is the thickness of the insulator and i its dielectric constant. By applying positive (negative)
bias electron (holes) can be induced in the sheet. The electron and hole densities depend on the Fermi
energies
3
n S =
8π
h 2 v
2
F
∞
E D
E − E D
1 + exp [(E − E F )/kT ]
dE
(13.10a)
p S =
8π
h 2 v
2
F
E D
−∞
−(E − E D )
1 + exp [−(E − E F )/kT ]
dE
(13.10b)
as visualized in Fig. 13.8. These relations cannot be inverted to obtain E F (n, p) analytically. The total
charge carrier density is ρ S = e( p S − n S ).
The Hall effect (Fig. 13.9) shows the expected ambipolar dependence according to (15.16) which
takes the form
3 Here we assume the linear dispersion for all thermally populated states.