13.1 Graphene and Related Materials
387
Fig. 13.6 Experimental bandstructure of graphene (on (0001) 6H-SiC) as determined from ARPES. a Energy distribution
of states as a function of momentum along principal directions in the Brillouin zone. The single-orbital tight-binding
model (13.2) with T = 2.82 eV is shown as solid lines. The Fermi level is shifted by 0.435 eV due to doping. b Constant
energy map of the states at binding energy corresponding to the Dirac energy E D ; the boundary of the Brillouin Zone
boundary is superimposed as dashed line. The arrow at the K-point indicates the directions over which the data in (e–h)
were acquired. c, d Constant energy maps at the Fermi energy E F = E D + 0.45 eV and E D − 1.5 eV, respectively. e–h
Experimental energy bands along the line through the K-point parallel to Γ –M direction as indicated in (b). The dashed
lines are an extrapolation of the lower bands below the Dirac crossing energy, which are observed not to pass through the
upper bands (above E D ), suggesting the kinked shape of the bands around E D . The sheet electron density is n S = 1.1,
1.5, 3.7, and 5.6 × 10 13 cm −2 for (e)–(h), respectively, due to increased doping upon potassium adsorption. Adapted
from [1254], reprinted with permission, ©2006, Springer Nature
13.1.3 Electrical Properties
The Shubnikov-de Haas (SdH) oscillations from a graphene sheet exhibit a behavior [1249]
1
B
=
4 e
h
1
n S
,
(13.3)
which corresponds to (15.39) for a two-dimensional electron system and a spin- and valley-degeneracy
1
of both two. The cyclotron mass has been determined from the temperature dependence of the SdH
oscillations to be proportional
2 to
√
n (Fig. 13.7). The cyclotron mass is generally related [1259] to the
area S(E) = πk
2 in k-space of the orbits at the Fermi energy via
m c =
2
2π
∂ S(E)
∂ E
(13.4)
With the linear dispersion (13.1) we can write (13.4) as
m c =
2
2π
2π E
2 v
2
F
=
E
v
2
F
.
(13.5)
For the linear energy dispersion (13.1) the number of states up to energy E F is (with a degeneracy of 4)
1 Each of the six valleys at the K-points is shared by three Brillouin zones; keep in mind that the K- and K’-points are
not equivalent.
2 In a parabolic dispersion as in (15.38), the cyclotron mass is independent of n.
387
Fig. 13.6 Experimental bandstructure of graphene (on (0001) 6H-SiC) as determined from ARPES. a Energy distribution
of states as a function of momentum along principal directions in the Brillouin zone. The single-orbital tight-binding
model (13.2) with T = 2.82 eV is shown as solid lines. The Fermi level is shifted by 0.435 eV due to doping. b Constant
energy map of the states at binding energy corresponding to the Dirac energy E D ; the boundary of the Brillouin Zone
boundary is superimposed as dashed line. The arrow at the K-point indicates the directions over which the data in (e–h)
were acquired. c, d Constant energy maps at the Fermi energy E F = E D + 0.45 eV and E D − 1.5 eV, respectively. e–h
Experimental energy bands along the line through the K-point parallel to Γ –M direction as indicated in (b). The dashed
lines are an extrapolation of the lower bands below the Dirac crossing energy, which are observed not to pass through the
upper bands (above E D ), suggesting the kinked shape of the bands around E D . The sheet electron density is n S = 1.1,
1.5, 3.7, and 5.6 × 10 13 cm −2 for (e)–(h), respectively, due to increased doping upon potassium adsorption. Adapted
from [1254], reprinted with permission, ©2006, Springer Nature
13.1.3 Electrical Properties
The Shubnikov-de Haas (SdH) oscillations from a graphene sheet exhibit a behavior [1249]
1
B
=
4 e
h
1
n S
,
(13.3)
which corresponds to (15.39) for a two-dimensional electron system and a spin- and valley-degeneracy
1
of both two. The cyclotron mass has been determined from the temperature dependence of the SdH
oscillations to be proportional
2 to
√
n (Fig. 13.7). The cyclotron mass is generally related [1259] to the
area S(E) = πk
2 in k-space of the orbits at the Fermi energy via
m c =
2
2π
∂ S(E)
∂ E
(13.4)
With the linear dispersion (13.1) we can write (13.4) as
m c =
2
2π
2π E
2 v
2
F
=
E
v
2
F
.
(13.5)
For the linear energy dispersion (13.1) the number of states up to energy E F is (with a degeneracy of 4)
1 Each of the six valleys at the K-points is shared by three Brillouin zones; keep in mind that the K- and K’-points are
not equivalent.
2 In a parabolic dispersion as in (15.38), the cyclotron mass is independent of n.