Advanced Solid State Physics

This is a differential equation for the velocityv. The velocity can be determined to be

v =

−eEt

m

+v 0

and is proportional to the timet, the positionxis given by

x =

−eEt^2

2 m

+v 0 t+x 0.

If the electrons are put into an magnetic field they move on a spiral. For a magnetic and an electric
field the electrons move perpendicular to both fields^10.

10.3 Drift-Diffusion

For the diffusive transport the electrons are also accelerated at the beginning like the ballistic electrons,
but after a timet 0 the electrons scatter and have a velocityv 0 which is randomly distributed. Hence
the average velocity after the scattering event is〈v 0 〉 = 0. Newton’s law for an electron in an electric
field is (again)

F = ma = −eE = m

dv

dt

,

and the velocityvcan be calculated as

v = −

eE(t−t 0 )

m

+v 0.

With the average velocity after a scattering event is being zero the drift velocityvd which is the
average velocity of the electrons is:

vd = 〈v〉 =

−eEτSC

m

=

−eEl

mvF

= −μE (105)

whereμis the mobility of the electrons andτSC = 〈t−t 0 〉is the average time between scattering
events. This means that the drift velocity is not proportional to the time any more only proportional
toτSCand the electric field, therefore also the current densityj = nqvis proportional to the electric
field which is Ohm’s law.
Without an electric field the average electron position in a magnetic field is zero, the electrons only
move on circular paths. However if the electrons are in an electric and magnetic field the average
electron position moves on a straight line at the Hall angle to the fields (in contradiction to the
ballistic transport where the electrons move perpendicular to the electric and magnetic field).
The described Drift-Diffusion above is valid for low frequencies, if the frequency of the electric field
is high the probability that the electrons scatter gets smaller. This is because the mean free path
is much longer than the distances the electrons travel before the electric field changes the direction.
Therefore the transport gets more ballistic.
In an insulator in an electric field no current is flowing, although the insulator gets polarized. For an
insulator in a changing electric field the polarisation gets changed and therefore a current is flowing
which is out of phase with respect to the electric field. In the high frequency regime the transport in
metals gets to be ballistic transport because of the unlikely electron scattering as mentioned above.
Therefore metals act like insulators, because the ballistic electrons move out of phase with respect to
the fast changing electric field which is a similar behaviour to an insulator.

(^10) In an electric and magnetic field electrons move with a drift velocityv= E×B
B^2