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### Goran Sljuka

### History

### History

### Hall Effect

Fractional Quantum Hall Effect

History of Hall Effect

- Hall Effect
- (Integer ) Quantum Hall Effect
- Fractional Hall Effect
- HEMT
- Topological Order

1878 The Hall Effect discovered by Edwin Hall, grad student at John Hopkins University, Baltimore, Maryland

1930 Lev Davidovich Landau(Nobel Prize 1962) for discivery of Landau levels which explain the integer quantum Hall effect

In 1980 Klaus von Klitzing discovered Integer QHE (Nobel Prize)

1982 FQHE experimentally discovered by Daniel C. Tsui and Horst L.Stormer

1985 Robert B. Laughlin showed that the electrons in a powerful magnetic field can condense to form a kind of quantum fluid related to the quantum fluids that occur in superconductivity and in liquid helium.

Consequence of the forces that are exerted on moving charges by electric and magnetic fields

Used to distinguish weather semiconductor is n(negative Hall voltage)or p type(positive Hall voltage)

Measure majority carrier concentration and mobility

Hall effect Summary

- Magnetic field applied perpendicular to direction of current –voltage developed in third direction
- Equilibrium: Magnetic force=Electrostatic force (Ey = vxBz )
- The Hall coefficient is defined as

RH= Ey/Bzjx

- The current density is

jx = vxNq RH=1/Nq

- RH increases linearly with magnetic field

Applications

- Split ring clamp-on sensor
- Analog multiplication
- Power sensing
- Position and motion sensing
- Automotive ignition and fuel injection
- Wheel rotation sensing

The Integer Quantum Hall Effect

- In 1980 Klaus von Klitzing discovered that the Hall resistance does not vary in linear fashion, but "stepwise" with the strength of the magnetic field
- The steps occur at resistance values that do not depend on the properties of the material but are given by a combination of fundamental physical constants divided by an integer. We say that the resistance is quantized

Quantum Hall effect requires

- 1. Two-dimensional electron gas
- 2. Very low temperature (< 4 K)
- 3. Very strong magnetic field (~ 10 Tesla)

The Integer Quantum Hall Effect

- Quantum mechanical version of Hall effect observed in two dimensional systems of MOSFETs

The Quantum Hall Effect

- Measurements where electron density varied in fixed magnetic field (resistance does not depend on material):

-plateau regions in the Hall resistivity where it remains constant as density is changed

- The the value of Hall resistivity in plateau region is given exactly by h/e2divided by a integer

The Quantum Hall Effect

- Hall resistivity h/ie2 at zero temperature is quantized and diagonal resistivity vanishes into plateau regions
- Facts established:

-differences between two samples of different size

-differences between two materials

- differences between different plateau

are smaller than 10^-10times the quantized value

The Quantum Hall Effect

The zeros and plateaux in the two components of the resistivity tensor are intimately connected and both can be understood in terms of the Landau levels (LLs) formed in a magnetic field.

.

Magnetic Quantisation and Landau Levels

- Cyclotron frequency of electrons wc=eB/m*.

The Integer Quantum Hall Effect

- Significant role of quantised Hall effect with development of transistors
- Hall effect observed in two dimensional systems of electrons at low temp. and strong magnetic field
- Hall resistance is quantised and is no longer increasing linearly with magnetic field
- (ν = 1, 2, 3, etc.).

The Integer Quantum Hall Effect

- 1985 Klaus von Klitcing received Nobel Prize for discovery that Quantum Hall Effect was exactly quantised
- Important points to note are:

-The value of resistance only depends on the fundamental constants of physics:e the electric charge and h Plank's constant.

-It is accurate to 1 part in 100,000,000.

The Integer Quantum Hall Effect

- The QHE can be used as primary a resistance standard, although 1 klitzing is a little large at 25,813 ohm!
- conventional value of von Klitzing constant Value 25 812.807 .
- Standard uncertainty (exact) .
- Relative standard uncertainty (exact) .

.

The Fractional Quantum Hall Effect

- Due to completely different physics discovered experimentally by Daniel C. Tsui and Horst L.Stormer on GaAs heterostructure
- Explained by Robert B. Laughlin in 1983 using novel quantum liquid phase that accounts for the interactions between electrons
- Directly observed fractional charges through measurement of quantum shot noise
- Influential in theories abut topological order

The Fractional Quantum Hall Effect

- Horst L. Störmer and Daniel C. Tsui made the discovery in 1982 in an experiment using extremely powerful magnetic fields and low temperatures.
- Robert B. Laughlin showed that the electrons in a powerful magnetic field can condense to form a kind of quantum fluid related to the quantum fluids that occur in superconductivity and in liquid helium.

FQHE

- The principle series of fractions that have been seen are listed below. They generally get weaker going from left to right and down the page:
- 1/3, 2/5, 3/7, 4/9, 5/11, 6/13, 7/15...
- 2/3, 3/5, 4/7, 5/9, 6/11, 7/13...
- 5/3, 8/5, 11/7, 14/9...
- 4/3, 7/5, 10/7, 13/9...
- 1/5, 2/9, 3/13...
- 2/7, 3/11...
- 1/7....

FQHE

- Plateaus in the Hall resistance traced as a function of magnetic field (or particle density)
- Resistance values extremely close to (h/e2)/f
- Filling factor, f, is an integer number
- e and h are fundamental constants of nature . the elementary charge of the electron and Planck.s constant.
- The filling factor is determined by the electrondensity and the magnetic flux density. (IQHE)

FQHE

- f is ratiof=N/N F between the number of electrons N and the number of magnetic fluxquanta NF= F/ F0 , where F is the magnetic flux through the plane and F0=h/e=4.1.10-15 Vs
- When f is an integer the electrons completely fill a corresponding number of the degenerate energy levels (Landau levels) formed in a two-dimensional electron gas under the influence of a magnetic field.

FQHE

- If the filling factor is a fraction, f=1/3 for instance, there is no energy gap in the independent electron model that defines the Landau levels.
- The gap of a few Kelvin that is important for the FQHE is caused by a strongly correlated motion of electrons and is induced by the magnetic field and the repulsive Coulomb interaction between the electrons.

Discovery of an anomalous quantum Hall effect

- Horst L. Störmer and Daniel C. Tsui were studying the Hall effect using very high quality gallium arsenide-based sample
- The purity of the samples was so high that the electrons could move ballistically, i.e. without scattering against impurity atoms, over comparatively long distances

FQHE vs IQHE

- As in IQHE, FQHE plateau are are formed when the Fermi energy lies in the gap of the density of states
- IQHE gaps are due to magnetic quantisation of single particle motion
- FQHE the gaps arise from collective motion of all the electrons in the system

FQHE vs IQHE

- arguments used to understand the integer effect were not applicable
- existence of quasiparticles carrying fractional charge
- For the state at filling factor 1/3 Laughlin found a many body wave function with lower energy than the single particle energy

Laughlin.s wave function - a theorist.s tour-de-force

- electron system condenses into a new type of quantum liquidwhen its density corresponds to .simple. fractional filling factors of the form f=1/m, where m is an odd integer; f=1/3 or 1/5 for example
- proposed an explicit many-electron wave function for describing the ground state of this quantum liquid of interacting electrons.

Laughlin.s wave function - a theorist.s tour-de-force

- Laughlinalso showed that an energy gap separates the excited states from the ground stateand that they contain .quasiparticles. of fractional charge ±e/m.
- To further discuss Laughlin’s novel ideas it is convenient to denote the position (xj,yj) of the j:th electron in a two-dimensional plane by a complex number zj= xj/yj.

Laughlin.s wave function - a theorist.s tour-de-force

- The wave function for Nelectrons can be written as a simple product over all differences between particlepositions

(zj- zk)

Ym(z1,z2,z3,.,zN) = (z1- z2)m (z1- z3)m (z2- z3)m . (zj- zk)m. (zN-1- zN)m

- The theory of 4He,which involves correlated many-particle wave functions (Jastrow functions)

HEMT

- HEMT transistors are built on material that was used in FQHE experiment
- HEMT - Wikipedia, the free encyclopedia

Phases of Matter

- Believed that all phases of matter are describedby Landau’s symmetry breaking theory.
- The Landau’s theory was developed for classical statistical systems which are described by positive probability distribution functions of infinite variables.
- . FQH states are described by their ground state wave functions which are complex functions of infinite variables.

Phases of Matter

- Liquid -random distribution-liquid remains the same as we displace it by an arbitrary distance. We say that a liquid has a continuous translation symmetry.
- phase transition -crystal -atoms organize into a regular array (a lattice ). A lattice remains unchanged only when we displace it by a particular distance, so a crystal has only discrete translation symmetry.

Phases of Matter(solid, gas,superfluid)

- The phase transition

liquid - continuous translation symmetry crystal - discrete translation symmetry change in symmetry -symmetry breaking.

- The essence of the difference between liquids and crystals is therefore that the atoms have different symmetries in the two phases.

The Quantum Numbers

The Landau symmetry-breaking description of a system

principal quantum number

azimuthal quantum number

magnetic quantum number

spin quantum number

- According to Landau’s theory - states of matters are characterized by their symmetries.

Topological Order

- Topological order is a new kind of order described by a new set of quantum numbers such as:

ground state degeneracy,

quasiparticle,

fractional statistic,

edge states,

topological entropy,

The topological order is new since it is not related to symmetries

Topological Order-Quantum Order

- all different FQH states have the same symmetry
- cannot be described by the Landau’s theory.
- The topological order- special case of quantum order

quantum order with a finite energy gap

References

- http://www.pha.jhu.edu/~qiuym/qhe/node1.html
- http://hyperphysics.phy-astr.gsu.edu/HBASE/magnetic/hall.html
- Advanced Lab - Hall Effect:Hall Effect Experiment
- http://www.warwick.ac.uk/~phsbm/qhe.htm
- Press Release: The 1998 Nobel Prize in Physics

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