Quantized resistance

Quantized Resistance Zhifan He, Huimin Yang Fudan University (China) April 9, Physics 141A

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General Resistance Hall Resistance Experiment of Quantum Hall Effect Theory of QHE Other Hall Effect

General Resistance

𝑈

𝑅= 𝐼 Attention: U and I are in the same direction

Today’s topic:

Hall Resistance

Attention: 𝑉𝐻 and I are not in the same direction 𝑉𝐻 𝐸𝑎 𝐵𝑣 𝑎 𝐵 𝐵 𝑅𝐻 = = = = = 𝐼 𝑗(𝑎𝑏) (𝑛𝑒𝑣)(𝑎𝑏) 𝑛𝑒𝑏 𝑛𝑠 𝑒

𝑛𝑠 is surface density in x-y plane

Resistance is continuous.

But when: 1. Change conductor into a special material 2. Temperature colds down to 4.2 K 3. Magnetic field rises up to 19.8 T

Interesting things happen….

Fixed B

Quantized Resistance! Fixed Ns

This phenomenon is called Integer Quantum Hall Effect

How to do this experiment? How to explain it?

ℎ = 25812.806𝛺 𝑒2

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Experiment of Quantum Hall Effect ·Date: 4th to 5th of February 1980 at around 2 a.m. ·Location: High Magnetic Field Laboratory in Grenoble ·Researcher: Klaus von Klitzing ·Finding: Integral Quantized Hall Effect ·Achievement: 1985 Nobel Prize in Physics

From Wikipedia

Sample and Methods ·Typical silicon MOSFET (The metal–oxide– semiconductor field-effect transistor) ·U(P − P)∝ 𝜌𝑥𝑥 and U(H − H)∝ 𝜌𝑥𝑦 (𝑅𝐻 ) ·A positive gate voltage increases the carrier density below the gate. ·Low temperatures (typically 4.2 K) and a strong magnetic field

From K. v. Klitzing, G. Dorda and M. Pepper, Phys. Rev. Lett. 45, 494 (1980)

The experimental curve ·The electrical resistance(𝜌𝑥𝑥 ) at B=0 and B=19.8T ·The Hall resistance(𝜌𝑥𝑦 ) ·Nice plateaus in the Hall resistance ·𝜌𝑥𝑦 = h/i𝑒 2 (h=Planck constant, e=elementary charge and i is the number of fully occupied Landau levels)

From K. v. Klitzing, G. Dorda and M. Pepper, Phys. Rev. Lett. 45, 494 (1980)

Explanation of the Quantum Hall Effect ·In the absence of magnetic field, the density of states in 2D is constant. ·Landau levels (LLs) formed in a magnetic field. ·The available states clump into Landau levels. ·When the Fermi energy lies in a gap between LLs, electrons cannot move to new states. http://www.youtube.com/watch?v=lVorIGNOtsg

The QHM and fine-structure constant “Realization of a Resistance Standard based on Fundamental Constants” “New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance” 𝜌𝑥𝑦 = h/i𝑒 2 h/𝑒 2  𝛼 −1 = (h/𝑒 2 )(2/𝜇0 c) = 137.036 ···

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Theory of QHE landau Level We focus on one electron in magnetic field. Use landau gauge: Magnetic vector potential :𝐴=(-By, 0, 0)

Schrodinger equation of electron in magnetic field 1 { [( pˆ x  eBy ) 2  pˆ y2 ]  eEy}   2m  

x ip x 1 e  ( y  yp ) Lx

1 mE yp  ( px  ) eB B

1 2

 ( px , N )  c ( N  )  eEy p  c 

e B m

m E   2 B

2

Behavior of electrons X direction: plane wave y direction: harmonic oscillator

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x ipx 1 e  ( y  yp ) Lx

1 mE yp  ( px  ) eB B

Question : when N=0, that is , every electron in y direction is in ground state, How many tracks in unit area?

Tip: In x direction, 𝑝𝑥 should be quantized so as to meet the periodic boundary condition. 2 px  Lx y p 

2 n p x  Lx 1 1 2

eB

p x 

eB

Lx

Number of tracks in area Lx*Ly:

N'

Ly y p

 Ly Lx

eB h

Number of tracks in unit area, which is surface density in x-y plane:

ns0

eB  ( N  0) h

Consider harmonic oscillator’s energy level in y direction: surface density should be:

eB ns  i h

𝒏𝒔 𝒊𝒔 𝒒𝒖𝒂𝒏𝒕𝒊𝒛𝒆𝒅! 𝐵 𝐵 1ℎ 𝑅𝐻 = = = 𝑛𝑠 𝑒 (𝑒𝐵 𝑖)𝑒 𝑖 𝑒 2 ℎ ℎ = 25812.806𝛺 𝑒2

Final Question: when fixed 𝑛𝑠 , how to explain the experiment in detail? How can quantized 𝑛𝑠 happen?

Start at point o, i=4

Increase B a little

When electrons change to local electrons, it needs lots of energy, so 𝑉𝐿 will increase rapidly to provide energy.

B increase more

𝐵 𝐵 1ℎ = = (1) B increase, 𝑅𝐻 = 𝑛𝑠 𝑒 (𝑒𝐵 3)𝑒 3 𝑒 2 ℎ 𝑅𝐻 will not change at all. Plateau (2) When local electrons go back to tracks. it doesn’t need to provide extra energy. So 𝑉𝐿 =0.

That’s all about Integer Quantum Hall Effect.

Other Hall Effect 1.Fractional Quantum Hall Effect 2.Anomalous Quantum Hall Effect No need external magnetic field! 3.Quantum Hall Effect in graphene Room temperature!(no need 4.2K) Next task: Achieve Quantum Hall Effect 1.at room temperature 2.Without external magnetic field 3. In common material Bright future :

Topological quantum computer

With much higher speed and much lower consumption

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Quantized resistance is quantized hall resistance. Quantized resistance means quantized surface density because of quantum effect in magnetic field.(landau level)

References 

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Guangjiong Ni,Advanced Quantum Mechanics.2nd ed. Fudan University. 2003 Quantum Hall Effect http://wenku.baidu.com/view/16b90fd228ea81c758f57840.html