Download Drude model and Magneto

Physics of Engineered and Nanostructured Materials (SS2011)
Sheet5 - 3.6.2011
Drude model and Magneto-Transport
a) In the framework of the Drude model, derive the motion equation of an electron of
~ and magnetic field B,
~ under the assumption
charge −e as a function of an electric field E
of a constant relaxation time τ . In the stationary state and for an electron density n, find
~ and the current density J,
~ for B
~ applied in
the relationship between the electric field E
~
the z-direction: B = (0,0,B). What are the resistance and conductivity tensors ρb and σ
b,
respectively?
b) What is the geometrical interpretation of the Hall angle θ defined by tan θ = µB, where
µ is the sample mobility?
c) In the case of a metallic sample of width w, length l and thickness d, with an electron
density n, calculate in the stationary state the value and sign of the Hall coefficient AH for
the case of a magnetic field applied in the z-direction and an externally applied electric field
Ex in the x-direction, again in the framework of the Drude model. The Hall coefficient is
related to the Hall resistivity RH (B) by:
RH (B) = AH
B
d
Solution:
a) Equation of motion:
~ − m~v + q~v × B
~ ,
m~v˙ = q E
τ
Stationary case: ~v˙ = 0
Current density: ~j = −en~v
Conductivity: σ0 = neµ
q = −e
Mobility: µ = eτ /m
Using the equations above for the current density, the conductivity and the mobility, the
stationary case of the equ. of motion results to:
~ × ~j)
~ = 1 (~j − µB
E
σ0
Now we can use Ohm’s law
~ = ρb~j
E
to obtain the resistivity tensor


1
+µB 0
1 
−µB
1
0
⇒ ρb =
σ0
0
0
1
conductivity: ρb−1 = σ
b

σ
b=
1
1+µ2 B 2
 µB
σ0  1+µ
2B2
0
µB
− 1+µ
2B2
1
1+µ2 B 2
0

0

0
1
~ and ~j not parallel (angle θ = tan µB between them). As shown in
b) Ohm’s law ⇒ E
~x
the figure below, the Hall angle is defined as the angle between the external electric field E
~
~
~
~
and the total field E = Ex + Ey . Here, Ey origins from the deflection of the electrons in the
magnetic field due to the Lorentz force.
  j0 
Ex
σ0
~x =  0  =  0 
=E
0
0

~ ext
E
Physics of Engineered and Nanostructured Materials (SS2011)
Sheet5 - 3.6.2011

 

j0
jx
~ = ρb~j = ρb  0  = 1 −µBj0 
E
σ0
0
0
~ x ,E)
~ = tan |Ey | = tan µB
⇒ ∠(E
|Ex |
c) Here:
~ = (0,0,B)
B
~ = (Ex ,Ey ,0)
E
~v = (vx ,vy ,0)
Equation of motion:
e ~
e
~ − ~v
~v˙ = − E
− ~v × B
m
m
τ
stationary state: vy = 0 (no charges can flow out of the sample), ~v˙ = 0
x-comp of equ of motion: ⇒ vx = −
y-comp of equ of motion: ⇒ 0 = −
with the cyclotron frequency ωc =
eτ
Ex
m
eτ
Ey + ωc τ vx
m
eB
m
⇒
Ey
= −ωc τ
Ex
Putting the results from above into the term for the Hall resistivity RH :
RH =
Uy
Ey w
Ey
Ex ωc τ
B
=
=−
= − eτ
=−
Ix
jx wd
envx d
en m Ex d
end
The Hall coefficient AH and the Hall resistivity RH are related by:
RH (B) = AH
⇒ AH = −
B
d
1
en
(always negative in the Drude model)
Additional:
Within a solid the mass in the cyclotron frequency equation above is replaced with the
effective mass tensor m∗ . Cyclotron resonance is therefore a useful technique to measure
effective mass and Fermi surface cross-section in solids. In a sufficiently high magnetic field
at low temperature in a relatively pure material
1
τ
> kB T
ωce >
~ωce
where t is the carrier scattering lifetime, kB is Boltzmann’s constant and T is temperature.
When these conditions are satisfied, an electron will complete its cyclotron orbit without
engaging in a collision, at which point it is said to be in a well-defined Landau level.