442
15 External Fields
Fig. 15.13 Magnetic field effect on excitons in WS 2 monolayer. a Normalized transmission spectra. b Energy position
of E σ + and E σ − of (circularly polarized) spectral lines of the 1–5 s exciton lines. c Calculated energy (E σ + + E σ − )/2,
showing the diamagnetic shift. Solid lines are model calculation with reduced exciton mass m ∗
r = 0.175 (and further
parameters), d Zeeman splitting E σ + − E σ − of the 1–3 s states together with linear fits (dashed line), g-factors are around
4, as labelled. Adapted from [1313], reprinted under Creative Commons Attribute (CC BY 4.0) license
Fig. 15.14 2D electron gas
in an external magnetic
field. a Allowed states in
k-space. b Density of states
(DOS) ρ versus energy.
Dashed line is
two-dimensional DOS
without magnetic field.
Thick vertical lines: δ-like
DOS without broadening,
curves: broadened DOS.
Based on [1123]
(a)
n=2
n=0
n=1
k y
E (0)
F
k x
(b)
E
E 1
E (0)
F
2D
E +h /2
1
c
h c
states (per unit area without spin degeneracy and without the degeneracy of the band extremum). In
reality, disorder effects lead to an inhomogeneous broadening of these peaks. The states in the tails of
the peaks correspond to states that are localized in real space.
Also, in a 2D system several physical properties exhibit an oscillatory behavior as a function of
Fermi level, i.e. with varying electron number, and as a function of the magnetic field at fixed Fermi
energy, i.e. at fixed electron number (Fig. 15.15).
15.2.7 Shubnikov-De Haas Oscillations
From the 2D density of states (per unit area including spin degeneracy) D
2D
(E) = m
∗
/π
2 the sheet
density of electrons n s can be expressed as a function of the Fermi level (at T = 0 K without spin
degeneracy)
n s =
m
∗
2π 2 E F .
(15.38)
Using (15.33) we thus find (without spin degeneracy, without valley degeneracy), that the period of
1/B is ∝ n s :
15 External Fields
Fig. 15.13 Magnetic field effect on excitons in WS 2 monolayer. a Normalized transmission spectra. b Energy position
of E σ + and E σ − of (circularly polarized) spectral lines of the 1–5 s exciton lines. c Calculated energy (E σ + + E σ − )/2,
showing the diamagnetic shift. Solid lines are model calculation with reduced exciton mass m ∗
r = 0.175 (and further
parameters), d Zeeman splitting E σ + − E σ − of the 1–3 s states together with linear fits (dashed line), g-factors are around
4, as labelled. Adapted from [1313], reprinted under Creative Commons Attribute (CC BY 4.0) license
Fig. 15.14 2D electron gas
in an external magnetic
field. a Allowed states in
k-space. b Density of states
(DOS) ρ versus energy.
Dashed line is
two-dimensional DOS
without magnetic field.
Thick vertical lines: δ-like
DOS without broadening,
curves: broadened DOS.
Based on [1123]
(a)
n=2
n=0
n=1
k y
E (0)
F
k x
(b)
E
E 1
E (0)
F
2D
E +h /2
1
c
h c
states (per unit area without spin degeneracy and without the degeneracy of the band extremum). In
reality, disorder effects lead to an inhomogeneous broadening of these peaks. The states in the tails of
the peaks correspond to states that are localized in real space.
Also, in a 2D system several physical properties exhibit an oscillatory behavior as a function of
Fermi level, i.e. with varying electron number, and as a function of the magnetic field at fixed Fermi
energy, i.e. at fixed electron number (Fig. 15.15).
15.2.7 Shubnikov-De Haas Oscillations
From the 2D density of states (per unit area including spin degeneracy) D
2D
(E) = m
∗
/π
2 the sheet
density of electrons n s can be expressed as a function of the Fermi level (at T = 0 K without spin
degeneracy)
n s =
m
∗
2π 2 E F .
(15.38)
Using (15.33) we thus find (without spin degeneracy, without valley degeneracy), that the period of
1/B is ∝ n s :