The document summarizes a lecture on the quantum Hall effect. It defines the quantum Hall effect as a phenomenon where the resistance of a quantum well system is quantized under low temperature and high magnetic field conditions. It then provides calculations to show that the quantum Hall resistance is quantized and equal to h/q^2, where h is Planck's constant and q is the elementary charge. Finally, it discusses how the quantization occurs due to the formation of discrete energy levels called Landau levels in the presence of a magnetic field.

1. Course: Quantum Electronics-II
Lecture: 7
Arpan Deyasi
Topic: Quantum Hall Effect
1
Arpan Deyasi, RCCIIT, India
6/25/2020

2. What is Q.H.E?
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Arpan Deyasi, RCCIIT, India
It is a phenomenon where
resistance of a quantum well system
is quantized under the condition of
low temperature and high magnetic field

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Arpan Deyasi, RCCIIT, India
2
2 2
*
( )
2
y z
E k k
m
= +

Calculation of Quantum Hall Resistance
Energy of quantum well system
No. of energy states per
energy interval in real space
2 2
1
( ) ( )
2
y z
N k k
κ
π
= +

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Arpan Deyasi, RCCIIT, India
Calculation of Quantum Hall Resistance
Density of states of any quantum well
system for a particular subband
*
2
( )
m
E
ρ
π
=


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Arpan Deyasi, RCCIIT, India
Calculation of Quantum Hall Resistance
K.E in magnetic field is quantized upto
Landau level because of cyclotron motion
Energy separation between two Landau level is c
ω

where
*
c
qB
m
ω =
cyclotron motion

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Arpan Deyasi, RCCIIT, India
Calculation of Quantum Hall Resistance
No. of energy states upto Landau level
( )
L c
N E
ρ ω
= 
*
2 *
L
m qB
N
m
π
= 

L
qB
N
π
=


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Arpan Deyasi, RCCIIT, India
Calculation of Quantum Hall Resistance
NS: no. of carriers upto Landau level
Quantum Hall Resistance QH
L
B
R
qN
=
QH
B
R
q qB
π
= ×

2
2
QH
h
R
q
=

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Arpan Deyasi, RCCIIT, India
Significance of Quantum Hall Resistance
Quantum Hall Resistance is constant
2
QH
h
R
iq
=

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Significance of Quantum Hall Resistance
Landau level below Fermi level are completely filled
and above are completely empty
Due to existence of impurity and disorder, sharp Landau
levels become broader
Landau levels with broader width are called Landau band

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Significance of Quantum Hall Resistance
Carriers at tail of Landau band are localized at lower
temperature, which don’t contribute to electrical conduction
Therefore, if Fermi level lies in localized state,
Hall resistance remains constant

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https://physics.stackexchange.com/questions/476184/quantum-hall-effect-
why-the-spikes-of-the-longitudinal-resistance-appear-every
References