Calculation of Quantum hall resistance

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?
6/25/2020 2Arpan 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

3. 6/25/2020 3Arpan Deyasi, RCCIIT, India
2
2 2
*
( )
2
y zE k k
m
= +

Calculation of Quantum Hall Resistance
Energy of quantum well system
No. of energy states per
energy interval in real space
2 21
( ) ( )
2
y zN k kκ
π
= +

4. 6/25/2020 4Arpan Deyasi, RCCIIT, India
Calculation of Quantum Hall Resistance
Density of states of any quantum well
system for a particular subband
*
2
( )
m
Eρ
π
=


5. 6/25/2020 5Arpan 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 cω
where
*c
qB
m
ω =
cyclotron motion

6. 6/25/2020 6Arpan Deyasi, RCCIIT, India
Calculation of Quantum Hall Resistance
No. of energy states upto Landau level
( )L cN Eρ ω= 
*
2 *L
m qB
N
mπ
= 

L
qB
N
π
=


7. 6/25/2020 7Arpan 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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Significance of Quantum Hall Resistance
Quantum Hall Resistance is constant
2QH
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

10. 6/25/2020 Arpan Deyasi, RCCIIT, India 10
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