vol III chap 12 sect 2
Previous: 12.1.Boson condensation.
12.2. Quantum Hall effects.¶
The Hall effect was discovered in 1879 by Edwin H. Hall (1855-1938). When an electric current flows through a conductor like a thin gold plate which is located in a magnetic field, this field exerts a transverse force on the carrier of moving charges, which tends to push them to one side of the conductor. The accumulation of charges on the sides of the conductor will balance the magnetic force, producing a measurable voltage between the two sides of the conductor. Such potential drop is at right angles both to the current and the magnetic field. Hall performed his experiments at room temperature and with moderate magnetic fields of less than one tesla (T). One century after, Hall effect has been observed at extremely low temperatures (only a few degrees from absolute zero, i.e. around -272°C) and very powerful magnetic fields (max approx. 30 T).
In what follows we refer to the 1985 Physics Nobel Prize awarded to Klaus von Klitzing “for the discovery of the quantized Hall effect” and to the 1998 Prize awarded to Robert B. Laughlin, Horst L. Störmer and Daniel C. Tsui “for their discovery of a new form of quantum fluid with fractionally charged excitations”. Next, we present the documents called WORK describing the main contributions made by the laureates and the title and subtitles of their Nobel Lectures. Afterwards, we present some information taken from the document Press release. Electrons in New Guises and insert some figures taken from the Nobel Lectures by von Klitzing and by Störmer. The appropriate references in MLA format are given at the end of the chapter.
1985 Physics Nobel Prize awarded to von Klitzing.
WORK: “If an electrical current flows lengthwise through a metal band and a magnetic field is placed against the surface of the band at a right angle, a charge arises diagonally in the band. Known as the Hall effect, it comes about because the movement of the electrons is deflected by the magnetic field. In 1980, Klaus von Klitzing discovered the quantum Hall effect in an interface between a metal and a semiconductor in a very clean material. In this effect, changes in the magnetic field result in changes in what is known as Hall conductance that vary in steps of whole-number multiples of a constant.”
NOBEL LECTURE: The Quantized Hall Effect by von Klitzing.
- Introduction
- Two- Dimensional Electron Gas
- Quantum Transport of a 2DEG in Strong Magnetic Fields
- Experimental Data
- Application of the Quantum Hall Effect in Metrology
Acknowledgements
MLA style: Klaus von Klitzing – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2023. Mon. 19 Jun 2023. https://www.nobelprize.org/prizes/physics/1985/klitzing/lecture/
1998 Physics Nobel Prize awarded to Laughlin.
WORK: “The Hall effect refers to the fact that if an electrical current flows lengthwise through a metal band and a magnetic field is placed against the surface of the band at a right angle, a charge arises diagonally in the band. In interfaces in certain materials a quantum Hall effect occurs. After Horst Störmer and Daniel Tsui discovered that changes in the magnetic field result in changes in Hall conductance that vary in steps that represent fractions of a constant, Robert Laughlin explained the phenomenon in 1983 with the formation of quasiparticles and a kind of quantum fluid.”
MLA style: Robert B. Laughlin – Facts. NobelPrize.org. Nobel Prize Outreach AB 2024. Sat. 27 Apr 2024. https://www.nobelprize.org/prizes/physics/1998/laughlin/facts/
NOBEL LECTURE: Fractional Quantization by Laughlin
SOLITONS
LOCALIZATION
FRACTIONAL QUANTUM HALL STATE
FRACTIONAL STATISTICS
REMARKS
MLA style: Robert B. Laughlin – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2023. Mon. 19 Jun 2023. https://www.nobelprize.org/prizes/physics/1998/laughlin/lecture/
1998 Physics Nobel Prize awarded to Störmer and Tsui.
WORK: “Hall effect refers to the fact that if an electrical current flows lengthwise through a metal band and a magnetic field is placed against the surface of the band at a right angle, a charge arises diagonally in the band. In interfaces in certain materials a quantum Hall effect occurs. Klaus von Klitzing discovered that changes in the magnetic field result in changes in what is known as Hall conductance that vary in steps of whole-number multiples of a constant. Subsequently, Horst Störmer and Daniel Tsui discovered in 1982 that there also are steps that represent fractions of the constant.”
MLA style: Horst L. Störmer – Facts. NobelPrize.org. Nobel Prize Outreach AB 2024. Sat. 27 Apr 2024. https://www.nobelprize.org/prizes/physics/1998/stormer/facts/
NOBEL LECTURE: The Fractional Quantum Hall Effect by Störmer.
INTRODUCTION
TWO-DIMENSIONAL ELECTRON SYSTEMS
MODULATION-DOPING
THE HALL EFFECT
THE INTEGRAL QUANTUM HALL EFFECT
THE FRACTIONAL QUANTUM HALL EFFECT
Discovery
Origin
Of Electrons and Flux Quanta
Composite Particles
Fermions and Bosons
Composite Particle Statistics
⅓ Fractional Quantum Hall State
The State at \(v = ½\)
All Those Other FQHE States
The Peculiar State at \(v = 5/2\)
CONCLUSIONS
EPILOGUE
BIBLIOGRAPHY
MLA style: Horst L. Störmer – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2023. Mon. 19 Jun 2023. https://www.nobelprize.org/prizes/physics/1998/stormer/lecture/
NOBEL LECTURE: Interplay of Disorder and Interaction in Two-Dimensional Electron Gas in Intense Magnetic Fields by Tsui.
PROLOGUE
TWO-DIMENSIONAL MAGNETO-TRANSPORT
QUANTUM PHASE TRANSITIONS IN IQHE
THE FQHE
THE MAGNETIC FIELD INDUCED CRYSTAL REGIME
ACKNOWLEDGEMENTS
MLA style: Daniel C. Tsui – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2024. Tue. 6 Feb 2024. https://www.nobelprize.org/prizes/physics/1998/tsui/lecture/
In 1980, Klaus von Klitzing discovered the quantum Hall effect in an interface between a metal and a semiconductor in a very clean material. At very low temperatures the electrons moved as in two dimensions only, and changes in the magnetic field resulted in changes in the Hall conductance that does not vary in linear fashion, but “stepwise”. The quantized values of the Hall resistance \(R_H\) were \((h/e2)/f\), where \(h\) is Planck’s constant, \(e\) is the elementary charge of the electron and \(f\) (the filling factor) is an integer number. According to von Klitzing, the fact that normal Ohmic resistance disappeared was an indication that the material became a sort of a superconductor. Next Figures 12.8 and 12.9 come from his NOBEL LECTURE The Quantized Hall Effect.
Figure 12.8. Experimental conditions for observing the quantized Hall effect.
Figure 12.9. Experimental results reported in von Klitzing Nobel Lecture; (\(h\) is Planck constant and \(e\) the charge of the electron).
In experiments with the electron gas realized during 1982-1983, Störmer and Tsui worked on more severe conditions of temperatures and magnetic fields (around -272°C and nearly 30 T). Although the electrons are most reluctant to condense (they are fermions) they combine with the “flux quanta” of the magnetic field and form composite particles that can condense (they become bosons). Störmer and Tsui found Hall plateaus at high magnetic field strengths corresponding to fractional values of the filling factor \(f\). This implied the existence of quasiparticles carrying fractional charge; for instance if \(f=1/3\) this would correspond to a fractional charge of \(e/3\).
In his 1998 Nobel Lecture \(The Fractional Quantum Hall Effect\) Störmer explains the main difference between IQHE (Integer Quantum Hall Effect) and FQHE (Fractional Quantum Hall Effect): “The IQHE can be understood solely on the basis of the quantized motion of individual 2D electrons in the presence of a magnetic field and random fluctuations of the interface potential which creates localized states. …. The origin of FQHE is interaction between electrons. It is therefore termed a many-particle effect or an electron correlation effect, since the charged electrons are avoiding each other by correlating their relative motion in an intricate manner.” (See Figure 12.10 taken from Störmer´s Nobel Lecture.)
(Source: MLA style: Horst L. Störmer – Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2023. Mon. 19 Jun 2023. https://www.nobelprize.org/prizes/physics/1998/stormer/lecture/)
Figure 12.10. Hall Resistance (\(R_H\)) showing many fractional values.
Although the electrons usually repel each other because they are fermions, according to Laughlin the reported fractional quantization was because in the presence of very strong magnetic fields the electron gas behaved as a new type of quantum fluid integrated by composite interacting fermions (the electrons) that behave as bosons and condense as fractionally charged “quasiparticles”. Laughlin, closed his Nobel lecture Fractional Quantization by saying: “Fractional quantum Hall quasiparticles are the elementary excitations of a distinct state of matter that cannot be deformed into noninteracting electrons without crossing a phase boundary. That means they are different from electrons in the only sensible way we have of defining different, and in particular are not adiabatic images of electrons the way quasiparticle excitations of metals and band insulators are.”