1. University of Washington
Quantum Hall Effect
Tun Sheng Tan
University of Washington
March
PHYS 486 Winter 2016

2. Hall Effect
In 1879, Edwin H. Hall discovered that a small transverse voltage appeared across a current-carrying thin metal strip in an applied magnetic field.
The discovery of the Hall effect enabled a direct measure of the carrier density.
The sign of Hall voltage showed that electrons are physically moving in an electric current.
Hall resistance eq

3. Electrons in a magnetic field experiences Lorentz force:
𝑚 𝑑 𝑣 𝑑𝑡 =−𝑒 𝐸 + 𝑣 𝑐 × 𝐵 +𝑚 (0− 𝑣 ) 𝜏
With 𝑗=𝜎𝐸=−𝑛𝑒𝑣, the resistivity is
𝜌= 𝑚 𝑛 𝑒 2 𝜏 𝐵 𝑛𝑒𝑐 − 𝐵 𝑛𝑒𝑐 𝑚 𝑛 𝑒 2 𝜏

4. Integer Quantum Hall Effect
In 1980, von Klitzing observed plateaus in Hall conductance in 2D electron gas quantized at integer multiples of 𝑒 2 ℎ . 𝜌 𝑥𝑥 =0 and 𝜌 𝑥𝑦 =𝑐𝑜𝑛𝑠𝑡
This discovery was surprising that von Klitzing was awarded the Nobel Prize.
Fractional QHE was discovered 2 years later but it is will not be discussed in this presentation.

5. Integer Quantum Hall Effect
Add picture of the model again
Measurement obtained by von Klitzing: Quantization of Hall resistivity.
𝑅 𝐻 =𝜌 𝑥𝑦 = Ω 𝑖

6. How to create 2D quantum system
The idea is to create a 2D system out of 3D system via “freezing ” one of the degree of freedom.
GaAs heterostructure has high mobility µ ∼ 107 cm2/Vs

7. N P P N
At low temperature (T < 4K) and low electron density, only 𝐸 0 sub-band is occupied.
the separation between electric subbands, which is of the order of 10 meV,

8. Landau Energy Level in a 3D system
In present of magnetic field, the Hamiltonian of an free electron is defined as 𝐻= 𝑝+𝑒𝐴 2 2𝑚 with B= 𝛻×𝐴=𝛻× −𝐵𝑦 =𝐵 𝑧 .
The wave function can be written as: 𝜓= 𝑒 𝑖 𝑘 𝑥 𝑥+ 𝑘 𝑧 𝑧 𝜙(𝑦)
The Hamiltonian could be written in the form:
− ℏ 2 2𝑚 𝜕 2 𝜕 𝑦 𝑚 ℏ 𝑘 𝑥 +𝑒𝐵𝑦 2 𝜙 𝑦 = 𝐸− ℏ 2 𝑘 𝑧 2 2𝑚 𝜙(𝑦)
This is a simple harmonic oscillator Hamiltonian.
The energy level is called Landau energy level:
𝐸= 𝑛 ℏ 𝜔 𝑐 + ℏ 2 𝑘 𝑧 2 2𝑚 ,𝑛∈ℕ, 𝜔 𝑐 = eB m
Degeneracy of each Landau level per unit area is 2 𝑒𝐵 ℎ
Each state occupies ℎ 𝑒𝐵 in real space corresponds to the area of a quantum flux.

9. Simple Explanation to QHE
Recall from statistical mechanics that a 2D system has a density of sate function g(E):
𝑔 𝐸 = 2𝜋𝑚 ℎ 2
So the number of states for Landau levels filled to the 𝑖 𝑡ℎ level is 𝑛=𝑖∗𝑔 𝐸 𝑑𝐸=𝑖 2eB h .
𝑅 𝐻 = 𝐵 𝑛𝑒 = ℎ 𝑖2 𝑒 2 = Ω 𝑖
Order of magnitude, what is dE, how to get R_H

10. Simple Explanation to QHE
The impurity lifts the degeneracy of Landau level. The Landau sub-bands broadens.
The impurity also creates two kinds of electronic states: localized and extended states. Localized states do not carry current. Extended states are states whose energy is close to the unperturbed states.
and the width given by Γ=2.3𝑚𝑒𝑉 𝑇 −1 × 𝐵 𝜇 .

11. Only the sequence of i: 8, 6, 4, 3, 2, 1 are observed. Why ?
In real experiment, there is spin-splitting. So, Landau level 𝑖→spin split Landay Level 𝑖.
𝑖= 𝑛ℎ 2𝑒𝐵 → 𝑛ℎ 𝑒𝐵 Only at high B that odd 𝑖 is observed.
At 1 Tesla, ℏ𝜔 𝑐 =2 𝑚𝑒𝑉and 𝑔 ∗ 𝜇𝐵=2.3∗ 10 −2 𝑚𝑒𝑉.
Measured Hall resistivity 𝑅 𝐻 = ℎ 𝑖 𝑒 2

12. Aharonov-Bohm Effect Even when B is zero, A is not zero.
Electron beams pick up a phase difference.

13. Laughlin Gauge Argument
Suppose that a strip of metal is in a magnetic field perpendicular to the surface. The edges are connected to two reservoirs. Assume that there is a adiabatically changing magnetic flux going through the hole.
Revise

14. The changing flux induces a current . 𝐼 = 𝑑𝐸 𝑑Φ
In each filled Landau level a single electron is adiabatically transported from one edge of the annulus to the other.
The phase picked up by an electron going around the loop is 𝛾 𝑛 = 𝑖2𝜋Φ ℎ 𝑒 . So, the only way for the Hamiltonian to remain the same is to have Φ to be integer multiple of ℎ 𝑒 .
For p-filled Landau levels, the total energy required is 𝐸=𝑝𝑒 𝑉 𝐻
Hence, the Hall resistivity 𝑅 𝐻 = 𝑉 𝐻 𝐼 = 1 𝑝𝑒 ℎ 𝑒 = h 𝑝 𝑒 2

15. Conclusion
Quantum Hall Effect requires 2D system, low temperature and pure material.
Integer QHE can be modeled as non-interacting electron gas in a magnetic field.
QHE also give a low energy approach to measure fine structure constant, 𝛼 via the relation 𝑅 𝐾 = ℎ 𝑒 2 = 𝜇 0 𝑐 2𝛼
QHE can be used to define the standard for resistivity.
Uncertainty 3.7*10^-9. Theoretical calculation 10^-9.

16. Reference
J.E Avron, D. Osadchy, and R. Seiler. A topological look at the quantum hall effect. Physics Today, 56(8):38-42, Aug 2003.
K.v. Klitzing, G. Dorda, and M. Pepper. New method for high-accuracy determination of the fine-structure constant based on quantized hall resistance. Phys. Rev. Lett., 45: , Aug 1980.
K. Klitzing. The quantized hall effect. Nobel Lecture, pages 1-31, December 1985.
Shi Guo. Introduction to Integer Quantum Hall Effect. Lecture Note, March 2016.
S. A. Parameswaran. Aspects of the Quantum Hall Effect. PhD Thesis. 2011

17. THE END
Simple questions ?

18. Gauss-Charles Bonnet Formula
Given a surface without a boundary, S:
1 2𝜋 𝑠 𝐾 𝑑𝐴 =2(1−𝑔)
This is a pure geometric property, not quantization required.
A generalization of the formula Chern-Bonnet formula, C= 1 2𝜋 𝑠 𝐾 𝑑𝐴
This formula no longer gives an even integer and g is not the number of holes anymore.

19. Topology and QHE
So far, the gauge invariant require that the pump go back to its original state but not the transported charge in different cycle must be the same.
In quantum mechanics, having the same state do not guarantee the same outcome. So, gauge invariant alone is not enough to justify that the same number of electrons is transferred in every cycle of the pump.
Laughlin proposed that if Hall resistivity is quantized, it should not depend on the details of geometry.
Thouless, Kohmoto, den Nijs and Nightingale showed that Hall conductance is related to topology.

20. Berry curvature
Michael Berry showed that a wave-function undergoing adiabatic evolution has a phase change called Berry’s phase, 𝛾 𝑛 = 𝑑𝑅∗ 𝐴 𝑛 𝑅 .
Berry connection, 𝐴 𝑛 𝑅 =𝑖 𝑛(𝑅) 𝛻 𝑅 𝑛(𝑅)
Berry curvature, Ω 𝑛 𝑅 = 𝛻 𝑅 × 𝐴 𝑛 𝑅

21. Hall Conductance and Topology
Recall that the Hall current 𝐼= 𝜕 𝜃 𝐻(𝜙,𝜃). Then 𝜓 𝐼 𝜓 =ℏΩ 𝜙 .
Using Kubo formula (describing linear-response Hall conductivity), 𝜎 𝐻 = 𝑒 2 ℎ 1 2𝜋 Ω 𝑑𝑆

22. Crash Course on Topology
Continuous deformation is allowed.
In topology, for an orientable surface, the genus, g is the number of “holes” of a surface.

23. 𝜎 𝐻 = 𝑒 2 ℎ 𝐶
Consider the failure of parallel transport around the little red loop.
The phase mismatch of parallel transport can be calculated as integral of the curvature over inside area or the outside area.
The integrals must be multiple of 2𝜋 so the difference between them is the Chern number. As the red loop gets smaller, the Chern number will becomes 2(1-g)
So for small deformation in Hamiltonian, the Chern number is constant. When a large deformation in Hamiltonian causes the ground state to cross over other eigenstates, Chern number is no longer well defined.

24. Conclusion
Quantum Hall effect is very interesting. There are many phenomena related to QHE. Fore example, Q-Spin-HE, FQHE, QH-edge states, etc. Too little time and too many things.
There are many alternative approaches to describe QHE some are more accurate than the other.
Topology helps to determine Hall resistivity in cases where the Landau energy is very complicated, and many more uses.

25. Hofstadter Model
Electrons on a 2D lattice acted on by homogeneous magnetic field.
Thermodynamic properties determined by magnetic flux, temperature, and chemical potential.
The only known way of computing Hall conductance is with Chern numbers.
The different color is described by different Chern integers.
This is an example of Frustration. Two different area scales in competition.
Quantum flux and superlattice.