Chapter 15
External Fields
Abstract The effects of external electric and magnetic fields on the electronic and optical properties
of bulk materials and quantum wells are discussed including the Stark effect and quantum-confined
Stark effect, the Hall effect and Quantized Hall Effects. The energy levels of the solid and its optical
and electronic properties depend on external electric and magnetic fields. In high magnetic fields and
at low temperatures the quantum Hall effects give evidence for new states of matter in many-body
systems.
15.1 Electric Fields
15.1.1 Bulk Material
The center-of-mass motion of the exciton is not influenced by a homogeneous electric field. The
Hamilton operator for the relative motion of an electron-hole pair of reduced mass μ along z in the
presence of an electric field E along the z direction is
ˆ
H = −
2
2μ
− e E z .
(15.1)
Here, the Coulomb interaction, leading to the formation of bound exciton states, is neglected. In the
plane perpendicular to the field (here the z direction) the solutions for the relative motion are just plane
waves.
In the electric field the bands are tilted (Fig. 15.1), i.e. there is no longer an overall band gap.
Accordingly, the wavefunctions are modified and have exponential tails in the energy gap.
After separation of the motion in the (x,y) plane the Schrödinger equation for the motion in the z
direction is
−
2
2μ
d
2
dz 2 − e E z − E z
φ(z) = 0 ,
(15.2)
which is of the type
d
2 f (ξ )
dξ 2 − ξ f (ξ ) = 0 ,
(15.3)
© Springer Nature Switzerland AG 2021
M. Grundmann, The Physics of Semiconductors, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-51569-0_15
429
External Fields
Abstract The effects of external electric and magnetic fields on the electronic and optical properties
of bulk materials and quantum wells are discussed including the Stark effect and quantum-confined
Stark effect, the Hall effect and Quantized Hall Effects. The energy levels of the solid and its optical
and electronic properties depend on external electric and magnetic fields. In high magnetic fields and
at low temperatures the quantum Hall effects give evidence for new states of matter in many-body
systems.
15.1 Electric Fields
15.1.1 Bulk Material
The center-of-mass motion of the exciton is not influenced by a homogeneous electric field. The
Hamilton operator for the relative motion of an electron-hole pair of reduced mass μ along z in the
presence of an electric field E along the z direction is
ˆ
H = −
2
2μ
− e E z .
(15.1)
Here, the Coulomb interaction, leading to the formation of bound exciton states, is neglected. In the
plane perpendicular to the field (here the z direction) the solutions for the relative motion are just plane
waves.
In the electric field the bands are tilted (Fig. 15.1), i.e. there is no longer an overall band gap.
Accordingly, the wavefunctions are modified and have exponential tails in the energy gap.
After separation of the motion in the (x,y) plane the Schrödinger equation for the motion in the z
direction is
−
2
2μ
d
2
dz 2 − e E z − E z
φ(z) = 0 ,
(15.2)
which is of the type
d
2 f (ξ )
dξ 2 − ξ f (ξ ) = 0 ,
(15.3)
© Springer Nature Switzerland AG 2021
M. Grundmann, The Physics of Semiconductors, Graduate Texts in Physics,
https://doi.org/10.1007/978-3-030-51569-0_15
429