15.2 Magnetic Fields
441
(cmp. Fig. 9.33) and its linear and quadratic Zeeman effect [1453] as visualized in Fig. 15.12. The
difference in transition energies for lines 4 and 1 depends linearly on the magnetic field and is equal to
the electron spin splitting. The quadratic Zeeman effect is also termed ‘diamagnetic shift’ and allows
a determination of the wavefunction size. In the present case, the electron Bohr radius at the neutral
phosphorus donor in Si was found to be a D = 1.33(5) nm (cmp. (7.22)) [1453].
15.2.5 Magnetic Field Effect on Excitons
The effect of an magnetic field on excitons is similar to that on the wave function at an impurity, except
that the electron and hole are effected simultaneously. The effect is two-fold: The Zeeman-like effect
on the spin states leads to a splitting of the exciton lines which become circularly polarized; for ‘weak’
fields the splitting is linear with the field. Additionally, the magnetic field leads to a reduction of exciton
size leading to a quadratic shift of the line center to larger energies, the so-called diamagnetic shift.
Calculations for various confinement geometries have been given in [1454]. The change of exciton
size leads also to modification of localization effects in disordered quantum wells [1455]. As example
the magnetic field dependence of quasi two-dimensional excitons in a WS 2 monolayer is shown in
Fig. 15.13 (cf. Sect. 13.2.4).
15.2.6 Energy Levels in a 2DEG
In a 2D electron gas (2DEG), e.g., in a quantum well or a potential well at a modulation-doped
heterointerface, a free motion in z is not possible and k z is quantized. The energy levels (for each 2D
subband) are only given by the cyclotron energy (Fig. 15.14a). The density of states is a sequence of
δ-like peaks (Fig. 15.14b). Each peak contributes (degeneracy ˆ
g of a Landau level) a total number of
ˆ
g =
e B
h
(15.37)
(a)
(b)
Fig. 15.12 a Schematic level scheme and transitions (and their polarization) of donor bound excitons at Si:P. b Experimental magnetic field dependent shift of photoluminescence transition energies (from about E = 1.150 eV at B = 0 T)
(symbols) together with fits for heavy (dashed lines) and light (solid lines) hole transitions. Adapted from [1453]
441
(cmp. Fig. 9.33) and its linear and quadratic Zeeman effect [1453] as visualized in Fig. 15.12. The
difference in transition energies for lines 4 and 1 depends linearly on the magnetic field and is equal to
the electron spin splitting. The quadratic Zeeman effect is also termed ‘diamagnetic shift’ and allows
a determination of the wavefunction size. In the present case, the electron Bohr radius at the neutral
phosphorus donor in Si was found to be a D = 1.33(5) nm (cmp. (7.22)) [1453].
15.2.5 Magnetic Field Effect on Excitons
The effect of an magnetic field on excitons is similar to that on the wave function at an impurity, except
that the electron and hole are effected simultaneously. The effect is two-fold: The Zeeman-like effect
on the spin states leads to a splitting of the exciton lines which become circularly polarized; for ‘weak’
fields the splitting is linear with the field. Additionally, the magnetic field leads to a reduction of exciton
size leading to a quadratic shift of the line center to larger energies, the so-called diamagnetic shift.
Calculations for various confinement geometries have been given in [1454]. The change of exciton
size leads also to modification of localization effects in disordered quantum wells [1455]. As example
the magnetic field dependence of quasi two-dimensional excitons in a WS 2 monolayer is shown in
Fig. 15.13 (cf. Sect. 13.2.4).
15.2.6 Energy Levels in a 2DEG
In a 2D electron gas (2DEG), e.g., in a quantum well or a potential well at a modulation-doped
heterointerface, a free motion in z is not possible and k z is quantized. The energy levels (for each 2D
subband) are only given by the cyclotron energy (Fig. 15.14a). The density of states is a sequence of
δ-like peaks (Fig. 15.14b). Each peak contributes (degeneracy ˆ
g of a Landau level) a total number of
ˆ
g =
e B
h
(15.37)
(a)
(b)
Fig. 15.12 a Schematic level scheme and transitions (and their polarization) of donor bound excitons at Si:P. b Experimental magnetic field dependent shift of photoluminescence transition energies (from about E = 1.150 eV at B = 0 T)
(symbols) together with fits for heavy (dashed lines) and light (solid lines) hole transitions. Adapted from [1453]