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QUANTUM HALL EFFECT
Classical Hall effect
B
1879 - E. Hall
v
e
F
𝑒𝑬𝑻 = 𝑒𝒗 × 𝑩
Carrier type, 𝑉𝐻 = 𝑏 𝑬𝑻 =
http://topocondmat.org/w3_pump_QHE/figures/hall_bar.svg
𝑏
JB,
𝑒𝑛
𝜎𝑥 = 𝑒𝑛𝜇𝐻 ,
𝜇𝐻 = 𝑟𝜇𝑒𝑓𝑓
r – Hall scattering factor
1
Classical Hall effect
The Drude model:
∗𝒗
𝑑𝒗
𝑚
𝑚∗
= −𝑒𝑬 − 𝑒𝒗 × 𝑩 −
𝑑𝑡
𝜏
Applied
Lorentz Scattering
electric field
force
𝐉 = −𝑒𝑛v
𝑱 = 𝜎𝑬
𝑒𝐵
𝑤𝐵 = ∗
𝑚
𝜎=
𝜎(𝐵=0)
1+𝑤𝐵 2 𝜏2
1
𝑤𝐵 𝜏
𝜎𝑥𝑥
−𝑤𝐵 𝜏
= −𝜎
1
𝑥𝑦
𝜎𝑥𝑦
𝜎𝑥𝑥
𝜎
𝐵=0
=
𝑒 2 𝑛𝜏
𝑚∗
2
Classical Hall effect
𝜌=
𝜌𝑥𝑥 =
𝜌𝑥𝑦
𝜎 −1
1
𝜎(𝐵=0)
=
1
𝜎(𝐵=0)
1
−𝑤𝐵 𝜏
𝑚∗
= 2
𝑒 𝑛𝜏
𝑒𝐵
𝑤𝐵 𝜏
𝐵
∗𝜏
𝑚
=
= 2
=
𝜎(𝐵=0)
𝑒 𝑛𝜏
𝑒𝑛
𝑚∗
𝜌𝑥𝑥
𝑤𝐵 𝜏
= −𝜌
1
𝑥𝑦
𝜌𝑥𝑦
𝜌𝑥𝑥
𝜌𝑥𝑦
𝜌𝑥𝑥
𝐵
3
Change experiment conditions
𝜌=
1
𝜎(𝐵=0)
1
−𝑤𝐵 𝜏
𝑤𝐵 𝜏
1
High 𝑤𝐵 𝜏:

High B
 >2T

High 𝜇
 material
 2DEG
 Low T (< 4K)
4
Quantum Hall Effect
Klitzing, 1980
(𝜌𝑥𝑦 )
(𝜌𝑥𝑥 )
M. Houssa, lecture series, KU Leuven.
5
Quantum Hall Effect
Why only 2
odd plateaus?
𝟐𝝅ℏ 𝟏
𝝆𝒙𝒚 = 𝟐
𝒆 𝝂
Why 5 is not
Plateaus
pronounced?
over a
range of
B
𝑷𝒆𝒂𝒌𝒔
𝑻𝒘𝒐 𝒑𝒆𝒂𝒌𝒔
𝝆𝒙𝒙 ≠ 0
𝝆𝒙𝒙 = 0
M. Houssa, lecture series, KU Leuven.
6
Semiclassical approach
Semiclassical approach:
v
ℎ
2𝜋𝑟 = 𝑛λ = 𝑛
𝑝
r
2
B
𝑛ℏ2
𝐸𝑛 =
𝑤𝐵
2
A bit wrong, but not bad for semiclassical approach
D.Tong
7
Quantum physics approach
Quantum mechanics approach:
1
𝐸𝑛 = ℏ𝑤𝐵 (n + )
2
Landau levels (LL)
E
n=0, 1, 2
DOS at each LL:
𝑁=
2𝑒𝐵
ℎ
Peaks spaced by ℏ𝑤𝐵 .
8
M. Houssa, lecture series, KU Leuven.
Quantum physics approach
Increasing B
𝑁=
2𝑒𝐵
ℎ
Energy
B=0
𝐸𝐹
ℏ𝑤𝐵
Density of states
Fermi level
between LL – no available states
crossing LL – states availabe
9
Quantum physics approach
Landau levels (LL)
E
Quantum mechanics approach:
1
𝐸𝑛 = ℏ𝑤𝐵 (n + )
2
n=0, 1, 2
Edge channel
Edge channel
Localised e
M. Houssa, lecture series, KU Leuven.
http://d29qn7q9z0j1p6.cloudfront.net/content/roypta/363/1834/2203/F8.large.jpg
http://i.stack.imgur.com/LT9cB.jpg
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𝝆𝒙𝒚 quantization
𝝆𝒙𝒚 =
M. Houssa, lecture series, KU Leuven.
𝟐𝝅ℏ 𝟏
𝒆𝟐 𝝂
11
𝝆𝒙𝒚 quantization
Edge states in
equilibrium with mL
mL
mR
𝑉𝐻
𝜌𝑥𝑦 =
𝐼
𝑒(𝜇𝑅 − 𝜇𝐿 )
𝑒𝑉𝐻 = 𝜇𝑅 − 𝜇𝐿
𝐼 = −𝑒𝑣𝑛 =
2𝜋ℏ
𝑣 - from a dispertion relation
n – sum over all filled states
Edge states in
equilibrium with mR
𝜌𝑥𝑦
2𝜋ℏ 1
= 2
𝑒 ν
ν - number of occupied LL
12
𝝆𝒙𝒚 quantization
Increasing B
𝑁=
Energy
B=0
2𝑒𝐵
ℎ
𝐸𝐹
ℏ𝑤𝐵
𝜌𝑥𝑦
2𝜋ℏ 1
= 2
𝑒 ν
Density of states
𝐵 ↑ → ν ↓→ 𝜌𝑥𝑦 ↑→ 𝑉𝐻 ↑
12
Origin of the plateaus
Plateaus
over a
range of
B
M. Houssa, lecture series, KU Leuven.
13
Origin of the plateaux
1
𝐸𝑛 = ℏ𝑤𝐵 (n + )
2
DOS at each LL: 𝑁 =
2𝑒𝐵
ℎ
Increasing B
Energy
B=0
𝐸𝐹
Density of states
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Density of
states
Origin of the plateaus
2𝑒𝐵
𝑁=
ℎ
Density of
states
Energy
Defects
Disorder
Energy
increases (within reason) – plateaus are more pronounced
no disorder – plateaus expected to disappear
David Tong, https://inspirehep.net/record/1127711/files/landau_levels.png
15
𝝆𝒙𝒙 oscillations
𝑷𝒆𝒂𝒌𝒔
𝝆𝒙𝒙 = 0
M. Houssa, lecture series, KU Leuven.
16
𝝆𝒙𝒙 oscillations
No backscattering on
the edges
Edge states in equilibrium with mL
Edge channel
Edge channel
mL
mR
Edge states in equilibrium with mR
Localised e
Energy
Increasing B
B=0
𝑒𝑉𝑥𝑥 = 𝜇𝑅 − 𝜇𝑅 = 0
𝐸𝐹
Density of states
Fermi level
between LL – no available states – no scattering - 𝝆𝒙𝒙 = 0
crossing LL – availabe bulk states between the edges - 𝝆𝒙𝒙 ≠ 0
17
Odd plateaus
𝑻𝒘𝒐 𝒑𝒆𝒂𝒌𝒔
M. Houssa, lecture series, KU Leuven.
𝝆𝒙𝒙 ≠ 0
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Zeeman splitting
ℏ𝑤𝐵 =
B=0
ℏ𝑒𝐵
𝑚∗
≫
𝐸𝑧 = 𝑔𝑠 𝜇𝐵 𝐵
B≠0
ν =2
Energy
𝑔𝑠 𝜇𝐵 𝐵
ν=3
𝐸𝐹
ℏ𝑤𝐵
↑
↑
↑
↑
ν =1
↑
↑
↑
↑
ν =2
𝐸𝐹
ν =1
Density of states
19
Zeeman splitting
𝑚∗
𝐸𝑧 = 𝑔𝑠 𝜇𝐵 𝐵
≫
ν =2
↑ ↑ ↑ ↑
ν=3
Energy
Energy
ν=3
𝑔𝑠 𝜇𝐵 𝐵
ℏ𝑤𝐵
↑ ↑ ↑ ↑
Density of states
Easy to jump
ν =2
↑ ↑ ↑ ↑
ℏ𝑤𝐵 =
ℏ𝑒𝐵
Density of states
Thermal excitation
Difficult to jump
Even number of LL occupied – pronounced plateaus
Odd number of LL occupied – not pronounced plateaus
20
Odd plateaus
𝑻𝒘𝒐 𝒑𝒆𝒂𝒌𝒔
M. Houssa, lecture series, KU Leuven.
𝝆𝒙𝒙 ≠ 0
21
Applications
 Standard of measuring resistivity: 𝜌𝑥𝑦 =
 Ratio of fundamental constants:
2𝜋ℏ1
𝑒2 ν
𝜋ℏ
𝑒2
 Quantum computers: error-free topologically protected states
22
Hall effects
Classical
Integer
QHE
Hall
effect
Fractional
QHE
Spin
QHE
23
Conclusions
 QHE – quantization of Hall voltage (transverse resistivity) as a
function of applied magnetic field at special conditions (2DEG, high
B and 𝜇, low T).
 Quantization of energy levels and modulation of the DOS by magnetic
field stands behind QHE.
 Due to finite size of the sample edge conducting states are formed. On
each edge current can propagate in only one direction. Thus
backscattering is only possible through the bulk of the sample when
there are available states. This leads to oscillations of the longitudinal
resistivity.
 Disorder is important.
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Thank you for your attention!
References
 Klitzing, K. V., Gerhard Dorda, and Michael Pepper. "New method for
high-accuracy determination of the fine-structure constant based on
quantized Hall resistance." Physical Review Letters 45.6 (1980): 494.
 The Quantum Hall Effect. TIFR Infosys Lectures. D.Tong.
Cambridge.
 The Quantum Hall Effect. Lecture series on NanoHub. S. Data.
Perdue University.
 The Quantum Hall Effect. Lectures series. M.Houssa. KU Leuven.
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Zeeman spliting
The gyromagnetic ratio of a particle or
system is the ratio of its magnetic
momentum in an atom to its angular
momentum.
Bohr magneton is a physical constant and
the natural unit for expressing the magnetic
moment of an electron caused by either
its orbital or spin angular momentum.
Classical Hall effect