Download Solid State 2 – Exercise 3

Solid State 2 – Exercise 4
1. Mobility, Diffusion, Fick law:
a) We define the electron mobility µ as the magnitude of velocity divided by the
electric field:
 | v | / E
In Drude’s model, find µ as a function of electron charge, mass and time between
collisions  coll .
b) Consider an electron gas with a finite gradient of density in the x direction.
n
0
x
Assume that in time  coll , electrons can cover a maximal distance   v   coll .
Show that (Fick’s law):
j  Dn
D  v 2  coll
where v is the electron velocity.
What approximation did you use ?
c) From a detailed kinetic calculation we find the exact diffusion coefficient is:
1
D  v 2  coll
3
For a free electron gas at temperature T,find µ as a function of D and the
temperature. What is this relation called ? Is this an important relation ?
d) Let’s look at electrons (you can do the holes yourself). The e current density
has two contributions:
- From diffusion due to density gradient
- From electron mobility due to electric field
dn
J e   n nc E  Dn c
Giving us:
dx
Prove that Je vanishes at thermal equilibrium.
n
for the free electron gas in which

Ek  2 k 2 / 2m , estimate its numerical value, and in particular determine its
dimensions. You may assume low temperatures.
2. Calculate explicitly the quantity e2
3. Plot the electrostatic potential across the depletion layer of a p-n junction, plot
the charge carrier density, and estimate the width of the depletion layer as
function of the temperature.
4. A Si p-n abrupt junction is created with Na=1018 cm-3 and Nd=1016 cm-3. Egap
=1.12eV, Nc(T) = 2.8E19, Pv(T) = 1.04E19. calculate Δφ, dn,dp, and maximum
electric field, Emax, in the junction at T= 300 K.
5. Degenerate Electron Gas (DEG):
a) What is a degenerate electron gas ? What is the connection between DEG
and a neutron star ?
b) Consider a 2 dimentional DEG. Show that
c) Now derive the degenerate einstein relation and show that
Where
is the density of states at the Fermi energy.
d) What happens to this relation in a 1D or a 3D gas ?
6. Derive and plot the temperature dependence of the saturation current.
7.