13.1 Graphene and Related Materials
389
Fig. 13.8 Band structure
of graphene with various
positions of the Fermi
energy E F in relation to the
Dirac energy E D . States
occupied with electrons are
shown with in bold
Fig. 13.9 Hall coefficient
(T = 10 K) for a graphene
sheet as a function of the
free carrier sheet density
ρ S /e = p S − n S (positive
values indicate p-type).
Data shown as solid lines
from [1249]
R H =
1
e
p S − n S
(n S + p S ) 2
(13.11)
for equal electron and hole mobilities.
4
From Hall effect measurements the mobility has been determined to be about 10
4 cm
2 /Vs, independent of temperature between 10 and 100 K, and the same for electrons and holes. This value, however, is
much smaller than the in-plane mobility of about 10
6 cm
2 /Vs at 4.2 K in high quality samples of highly
ordered pyrolithic graphite (HOPG) [1260]. In suspended graphene a mobility of 2.3 × 10
5 cm
2 /Vs has
been found, limited by finite sample size [1261]. Thus in graphene layers on a solid surface extrinsic
effects such as charge traps, interfacial phonons, ripples or fabrication residue seem to limit the carrier
mobility.
The QHE has been observed in graphene [1249], reportedly up at room temperature [1262]. The
plateaus (4e
2
/ h)(n + 1/2) correspond to unusual half-integer filling, the first plateau occurring at
2e
2
/ h, as has been also suggested from theory, and related to the ‘pseudospin’. Another consequence
of the Dirac-like behavior of the fermions in graphene is the presence of finite maximum resistivity
ρ max = h/4e
2
= 6.45 k even at low temperature and E F = E D . It is due to the fact that localization
effects, leading to insulating behavior, are strongly suppressed. Then each carrier keeps a mean free
path in the order of its Fermi wavelength.
The Klein paradox [1263, 1264], the efficient tunneling of Dirac particles through high and thick
barriers seems accessible in transport experiments with graphene [1265].
4 Including sign, μ h = −μ e . For T = 0 and E F = E D , n S = p S = 0 and thus 1/R H should be zero. For finite temperatures
there is always n S > 0 and p S > 0, even for E F = E D . Thus 1/R H ∝ 1/( p S − n S ) diverges at ρ S = e( p S − n S ) = 0
with a change of sign.
389
Fig. 13.8 Band structure
of graphene with various
positions of the Fermi
energy E F in relation to the
Dirac energy E D . States
occupied with electrons are
shown with in bold
Fig. 13.9 Hall coefficient
(T = 10 K) for a graphene
sheet as a function of the
free carrier sheet density
ρ S /e = p S − n S (positive
values indicate p-type).
Data shown as solid lines
from [1249]
R H =
1
e
p S − n S
(n S + p S ) 2
(13.11)
for equal electron and hole mobilities.
4
From Hall effect measurements the mobility has been determined to be about 10
4 cm
2 /Vs, independent of temperature between 10 and 100 K, and the same for electrons and holes. This value, however, is
much smaller than the in-plane mobility of about 10
6 cm
2 /Vs at 4.2 K in high quality samples of highly
ordered pyrolithic graphite (HOPG) [1260]. In suspended graphene a mobility of 2.3 × 10
5 cm
2 /Vs has
been found, limited by finite sample size [1261]. Thus in graphene layers on a solid surface extrinsic
effects such as charge traps, interfacial phonons, ripples or fabrication residue seem to limit the carrier
mobility.
The QHE has been observed in graphene [1249], reportedly up at room temperature [1262]. The
plateaus (4e
2
/ h)(n + 1/2) correspond to unusual half-integer filling, the first plateau occurring at
2e
2
/ h, as has been also suggested from theory, and related to the ‘pseudospin’. Another consequence
of the Dirac-like behavior of the fermions in graphene is the presence of finite maximum resistivity
ρ max = h/4e
2
= 6.45 k even at low temperature and E F = E D . It is due to the fact that localization
effects, leading to insulating behavior, are strongly suppressed. Then each carrier keeps a mean free
path in the order of its Fermi wavelength.
The Klein paradox [1263, 1264], the efficient tunneling of Dirac particles through high and thick
barriers seems accessible in transport experiments with graphene [1265].
4 Including sign, μ h = −μ e . For T = 0 and E F = E D , n S = p S = 0 and thus 1/R H should be zero. For finite temperatures
there is always n S > 0 and p S > 0, even for E F = E D . Thus 1/R H ∝ 1/( p S − n S ) diverges at ρ S = e( p S − n S ) = 0
with a change of sign.