Download I HALL EFFECT For a diffusive metal the Lorentz force law

I
I.
HALL EFFECT
HALL EFFECT
Classical Hall effect in a diffusive metal
For a diffusive metal the Lorentz force law can be written as
~F = −e ~E +~vd × ~B = m~vd ,
τ
(1)
where τ is the scattering time and ~vd the drift velocity.
Classical Hall resistivity: ρxy
The resistivity is defined by ~E = ρ ~j where j is the current density, this reads:
Ex
ρxx ρxy
jx
=
Ey
ρyx ρyy
jy
(2)
We substitute jx = σ0 Ex = −nevx . Using above definition we get:
eτ
ne2 τ
(Ex + vy Bz ) ,
jx = σ0 Ex = −envx = −ne − (Ex + vy Bz ) = −
m
m
(3)
which we can write as follows:
2
σ0 Ex = jx + nem B τvy
= jx + ωc τ jy
σ0 Ey = jy − ωc τ jx
In the last equation we set jy = 0, use j = σ0 E =
ρxy =
ne2 τ
m E
(4)
(5)
(6)
and obtain:
Ey
eBτ
eBτm
B
ωc τ
=−
=−
= − =: RH B,
=−
jx
σ0
mσ0
mne2 τ
ne
(7)
1
the Hall constant.
with RH = − ne
(Integer) Quantum Hall effect
In a 2D electron gas (2DEG) at low temperatures under the influence of a (strong) magnetic field the k-states are quantized
in what is known as Landau states or Landau circles. As such the energies are quantized as well: En = h̄ωc n + 12 . This
quantization leads to periodic oscillation of conductivity (Shubnikov-de Haas effect) or magnetic moment (De Haas-van Alphen
effect) in 1/B
From the De Haas-van Alphen effect we obtained the result for the density of states in the quantized Landau states is the
difference in areas (we are in 2D now): ∆S = (Sn − Sn−1 ) = 2πeB
h̄ which yields the density of states for the (magnetic) Landau
levels: D = (2πeB/h̄)(L/2π)2 =: ρB. The fermi level has a strong B-field dependence in this situation.
Lets assume the Landau level s ∈ N+ is fully occupied. If there are electrons in the next higher level s + 1 the Fermi level is
situated in this level. If the B-field is increased the electrons will move to the lower energy level since D ∝ B. As soon as all
remaining electrons of the s + 1 level are depleted the Fermi level will suddenly drop to the s level. This explains the oscillating
behavior. In analogy to the classical Hall constant we now define for the QHE:
rH = −
B
h 1
=− 2 ,
νNL e
e ν
(8)
with ν a positive integer number. Whilst this describes the periodicity it does not describe the characteristic plateaus. Those can
be understood after considering the transport mechanism. The presence of a magnetic field forces electrons into circular orbits
(cyclotron orbits). Near the boundaries of the solid those circles become partial circles, called skipping orbits.This confinement
leads to an upward shift in kinetic energy as described by the uncertainty principle and the energy levels at the edges can thus
cross the Fermi-level (we are still at low temperatures so the Fermi level is sharp). This edge states with a metallic character
are called edge channels, transport in this channels is quasi ballistic, even in the presence of impurities, since Lorentz force will
bend the trajectory of scattered electrons forward.
1
I
HALL EFFECT
Y
µL
BZ
µR
X
S
Figure 1. Formation of edge channels and schematic illustration of skipping orbits. In presence of a scatterer S the quasi-ballistic transport is
preserved.
The two edge channels are isolated from each other (in absence of thermal excitation in the sample that would allow electrons
in the bulk to cross the Fermi edge). If the sample is contacted left and right and a voltage V is applied across the contacts one
creates a difference in chemical potential: µL − µR = eV . The electric current difference of the left and right (1-dimensional)
channel can be expressed as:
nc
I=
∑
Z µL
eD1D
n (E)vn (E)dE
(9)
n=1 µR
when we use D(E) = 1/(2π)(dkx /dEn ) and vn = 1/h̄(dEn /dkx ) we can obtain a well known result:
e2
e
In = (µL − µR ) = V,
h
h
(10)
where we find, precisely, the value for the Hall conductance contributed by each channel to be e2 /h.
For the discovery of the exact quantization of the Hall conductance the Nobel Prize in physics was awarded to its discoverer
Klaus von Klitzing in 1985 [1].
Fractional QHE
In semiconductor heterostructures with high mobility smaller plateaus in the Hall resistance are observed. Additional plateaus
are formed at fractions 1/3, 2/5 2/3 etc of the Klitzing constant RK = eh2 . For this discovery the Nobel prize 1998 in physics
was awarded to H.L. Störmer, Daniel C. Tsui and Robert B. Laughlin [2].
[1] K. v. Klitzing, G. Dorda, and M. Pepper. New method for high-accuracy determination of the fine-structure constant based on quantized
hall resistance. Physical Review Letters, 45(6):494–497, August 1980.
[2] D. C. Tsui, H. L. Stormer, and A. C. Gossard. Two-dimensional magnetotransport in the extreme quantum limit. Physical Review Letters,
48(22):1559–1562, May 1982.
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