444
15 External Fields
Fig. 15.16 Shubnikov-de
Haas oscillations at a
modulation-doped
(Al,Ga)As/GaAs
heterostructure with a
2DEG,
n = 1.7 × 10 17 cm −2 and
μ = 11 400 cm 2 /Vs. Data
from [1457]
Fig. 15.17 Shubnikov-de
Haas oscillations of a
2DEG at the (100) surface
of p-type silicon
(100 cm) at a magnetic
field of B = 3.3 T and
T = 1.34 K. The inset
shows schematically the
contact geometry. Data
from [1458]
0
40
60
80
100
20
silicon
oxide
n+
n+
15.2.8 Quantum Hall Effect
In high magnetic fields, at low temperatures and for high-mobility, 2D electron gases exhibit a deviation
from the classical behavior. We recall that the classical Hall effect (i.e. considering the Lorentz force,
classical Drude theory), the generation of a field E y perpendicular to a current flow j x (cf. Sect. 15.2.1),
was described with the conductivity tensor σ (here, for the (x, y)-plane only)
σ =
σ 0
1 + ω 2
c τ 2
1 ω c τ
−ω c τ 1
(15.40a)
σ xx = σ 0
1
1 + ω 2
c τ 2 → 0
(15.40b)
σ xy = σ 0
ω c τ
1 + ω 2
c τ 2 →
n e
B
,
(15.40c)
where σ 0 is the zero-field conductivity σ 0 = ne
2
τ/m
∗ (8.5). The arrows denote the limit for ω c τ → ∞,
i.e. large fields. The resistivity tensor ρ = σ
−1 is given by
ρ =
ρ xx ρ xy
−ρ xy ρ xx
(15.41a)
ρ xx =
σ xx
σ 2
xx + σ 2
xy
→ 0
(15.41b)