Quantized Resistance

Zhifan He, Huimin Yang

Fudan University (China)

April 9, Physics 141A

Outline

General Resistance

Hall Resistance

Experiment of Quantum Hall Effect

Theory of QHE

Other Hall Effect

General Resistance

𝑅 =𝑈

𝐼

Attention: U and I are in the same direction

Today’s topic: Hall Resistance

𝑅𝐻 =𝑉𝐻

𝐼=

𝐸𝑎

𝑗(𝑎𝑏)=

𝐵𝑣 𝑎

(𝑛𝑒𝑣)(𝑎𝑏)=

𝐵

𝑛𝑒𝑏=

𝐵

𝑛𝑠𝑒

Attention: 𝑉𝐻 and I are not in the same direction

𝑛𝑠 is surface density in x-y plane

But when: 1. Change conductor into a special material

3. Magnetic field rises up to 19.8 T

Interesting things happen….

Resistance is continuous.

2. Temperature colds down to 4.2 K

How to explain it?

How to do this experiment?

Fixed B

Fixed Ns

Quantized Resistance!

This phenomenon is called Integer Quantum Hall Effect

ℎ

𝑒2 = 25812.806𝛺

Outline

General Resistance

Hall Resistance

Experiment of Quantum Hall Effect

Theory of QHE

Other Hall Effect

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

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

·Typical silicon MOSFET (The metal–oxide–semiconductor field-effect transistor) ·U(P − P)∝ 𝜌𝑥𝑥and 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

Sample and Methods

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

·The electrical resistance(𝜌𝑥𝑥) at B=0 and B=19.8T ·The Hall resistance(𝜌𝑥𝑦)

·Nice plateaus in the Hall resistance ·𝜌𝑥𝑦= h/i𝑒2

(h=Planck constant, e=elementary charge and i is the number of fully occupied Landau levels)

The experimental curve

Explanation of the Quantum Hall Effect

http://www.youtube.com/watch?v=lVorIGNOtsg

· 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.

The QHM and fine-structure constant

“Realization of a Resistance Standard based on Fundamental Constants”

h/𝑒2 𝛼−1 = (h/𝑒2)(2/𝜇0c) = 137.036 · · ·

“New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance”

𝜌𝑥𝑦= h/i𝑒2

Outline

General Resistance

Hall Resistance

Experiment of Quantum Hall Effect

Theory of QHE

Other Hall Effect

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

2 21ˆ ˆ{ [( ) ] }

2x yp eBy p eEy

m

1( )

x

xip

p

x

e y yL

1( )p x

mEy p

eB B

21

( , ) ( )2 2

x c p

m Ep N N eEy

B

c

eB

m

Behavior of electrons X direction: plane wave

y direction: harmonic oscillator

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

1( )

x

xip

p

x

e y yL

1( )p x

mEy p

eB B

Tip:

In x direction, 𝑝𝑥 should be quantized so as to meet the periodic boundary condition.

2x

x

p nL

2x

x

pL

1 1 2p x

x

y peB eB L

'y

y x

p

L eBN L L

y h

Number of tracks in area Lx*Ly:

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

0( 0)s

eBn N

h

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

s

eBn i

h

𝒏𝒔 𝒊𝒔 𝒒𝒖𝒂𝒏𝒕𝒊𝒛𝒆𝒅!

𝑅𝐻 =𝐵

𝑛𝑠𝑒=

𝐵

(𝑒𝐵ℎ

𝑖)𝑒=

1

𝑖

ℎ

𝑒2

ℎ

𝑒2 = 25812.806𝛺

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

Start at point o, i=4

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

Increase B a little

𝑅𝐻 =𝐵

𝑛𝑠𝑒=

𝐵

(𝑒𝐵ℎ

3)𝑒=

1

3

ℎ

𝑒2 (1) B increase,

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.

B increase more

That’s all about Integer Quantum Hall Effect.

Other Hall Effect 1.Fractional Quantum Hall Effect

2.Anomalous Quantum Hall Effect

3.Quantum Hall Effect in graphene

Next task: Achieve Quantum Hall Effect 1.at room temperature 2.Without external magnetic field 3. In common material

No need external magnetic field!

Room temperature!(no need 4.2K)

Bright future : Topological quantum computer

With much higher speed and much lower consumption

Conclusions

Quantized resistance is quantized hall resistance.

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

References

Guangjiong Ni,Advanced Quantum Mechanics.2nd ed. Fudan University. 2003

Quantum Hall Effect

http://wenku.baidu.com/view/16b90fd228ea81c758f57840.html