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

Transcription

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

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

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

4 Today s topic: Hall Resistance Attention: V H and I are not in the same direction R H = V H I = Ea j(ab) = Bv a (nev)(ab) = B neb = B n s e n s is surface density in x-y plane

5 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.

6 Fixed B Quantized Resistance! Fixed Ns h e 2 = Ω This phenomenon is called Integer Quantum Hall Effect How to do this experiment? How to explain it?

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

8 Experiment of Quantum Hall Effect Date: 4 th to 5 th 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

9 Sample and Methods Typical silicon MOSFET (The metal oxide semiconductor field-effect transistor) U(P P) ρ xx and U(H H) ρ xy (R H ) 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)

10 The experimental curve The electrical resistance(ρ xx ) at B=0 and B=19.8T The Hall resistance(ρ xy ) Nice plateaus in the Hall resistance ρ xy = 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)

11 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.

12 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 ρ xy = h/ie 2 h/e 2 α 1 = (h/e 2 )(2/μ 0 c) =

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

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

15 Schrodinger equation of electron in magnetic field 1 2m ˆ ˆ 2 2 { [( px eby) py] eey} 1 L x ip x x e ( y y ) y p p 1 me ( px ) eb B 1 me ( px, N) c( N ) eey p 2 2 B 2 c e B m

16 Behavior of electrons X direction: plane wave y direction: harmonic oscillator 1 L x ip x x e ( y y ) p y p 1 me ( px ) eb B Question : when N=0, that is, every electron in y direction is in ground state, How many tracks in unit area?

17 Tip: In x direction, p x should be quantized so as to meet the periodic boundary condition. p x 2 L x n p x 2 L yp px eb eb L Number of tracks in area Lx*Ly: x x N ' L y y p L L y x eb h Number of tracks in unit area, which is surface density in x-y plane: eb ns ( N 0) 0 h

18 Consider harmonic oscillator s energy level in y direction: surface density should be: n s eb h i n s is quantized! R H = B n s e = B ( eb h i)e = 1 i h e 2 h e 2 = Ω

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

20 Start at point o, i=4 Increase B a little When electrons change to local electrons, it needs lots of energy, so V L will increase rapidly to provide energy.

21 B increase more (1) B increase, R H = B n s e = B ( eb h 3)e = 1 3 h e 2 R H will not change at all. Plateau (2) When local electrons go back to tracks. it doesn t need to provide extra energy. So V L =0. That s all about Integer Quantum Hall Effect.

22 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

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

24 References Guangjiong Ni,Advanced Quantum Mechanics.2 nd ed. Fudan University Quantum Hall Effect