9 Magnetism and Response to Electric and Magnetic Fields
9.1 Introduction: Electric and Magnetic Fields
Physical quantities in solids can be calculated the following way:
Solve the Schroedinger equation→get the dispersion relation→calculate the density of states→
use this to get the Gibbs free energy→take certain derivatives to get quantities as the specific heat,
pressure, piezoelectric constants, piezomagnetic constants, etc.
The main problem seems to be calculating the microscopic states, as anything else is a straight forward
calculation. However, to get the microscopic states, one uses different types of models. Till now, we
often used the free electron model, and the electric field and the magnetic field were expected to be
zero. Because of this, the piezoelectric and the piezomagnetic constants turned out to be zero too,
when calculating the free energy.
As we want to know, how a material responses to an electric or a magnetic field, the simplifications
mentioned above, have to be corrected.
9.2 Magnetic Fields
When a charged particle (in most cases a free electron) moves through a magnetic field, there is
a force on this particle, perpendicular to the velocity. Remember that this force can change the
direction of the electron, but the magnetic field does no mechanical work, because of the electron
moving perpendicular to the force.
F=−ev×B (62)evBz=
mv^2
R(63)
v=wcR (64)wc=eBz
m(65)
withwcthe angular frequency or cyclotron frequency.
When the particle is moving in a circle around a located position, it is moving around the origin in
k-space as shown in fig. 58.
Usually when a charge is accelerated (which it is, when moving on a circle), it emits electromagnetic
radiation. For solids, a similar thing would be to put a metal into a magnetic field. The electrons
at the fermi surface are moving quite fast and there is also a perpendicular force on them when the
metal is in the field, but they are not emitting radiation. The reason for this is that they simply can’t
get into a state with lower kinetic energy (all filled up) like a free electron does.
The problem of getting the magnetic field into the Schroedinger equation is that the force depends on
the velocity as shown below.
F=q(E+v×B) (66)