Introduction to Quantum Hall Effects

Home , Landau quantization

MarkMark OliverOliver GoerbigGoerbig

EMFL Summer School “Science in High Magnetic Fields”, Arles, 29/09/2018 Reminder: ClassicalClassical Hall effect (1879)

(Quantum) Hall system: 2D electrons in a (perp.) Magnetic field B

Hall resistance: obtained from stationary equation of motion (Drude model)

Resistivity and conductivity

From Drude model: Cyclotron frequency :

Drude conductivity :

Resistivity and conductivity

From Drude model: Cyclotron frequency :

Drude conductivity :

Ohm’s law: current density:

→ resistivity/conductivity tensor:

Resistivity and conductivity

From Drude model: Cyclotron frequency :

Drude conductivity :

Ohm’s law: current density:

→ resistivity/conductivity tensor:

Link with mobility :

Hall resistivity: Shubnikov–de-Haas effect (1930) see A. Pourret’s class

Oscillations in longitudinallongitudinal resistance when

→ Einstein relation:

→ Landau quantization (into levels ): Quantum Hall effect (1980)

Signature of QHE:

→ plateau in

→ vanishing

quantized in terms of universal constant

→ use for metrology (resistance standard) Quantum Hall effect (1980)

Signature of QHE:

→ plateau in

→ vanishing

v. Klitzing (1985) Integral quantum Hall effect (IQHE, 1980)

Fractional quantum Hall effect (FQHE, 1982) Laughlin; Störmer, Tsui (1998) Ways to understand the QHE

Ways to understand the QHE

SpoilerSpoiler AlertAlert

Ways to understand the QHE

● LandauLandau quantizationquantization → energy of charged particles in a magnetic field is quantized into highly degenerate LandauLandau levelslevels

Ways to understand the QHE

● LandauLandau quantizationquantization → energy of charged particles in a magnetic field is quantized into highly degenerate LandauLandau levelslevels ● FromFrom energyenergy quantizationquantization toto conductanceconductance quantizationquantization → every filled Landau level contributes to the conductance → sample edges are conductanceconductance channelschannels → additional electrons in partially filled Landau levels get localizedlocalized byby impuritiesimpurities and do not contribute to the electronic transport References and further reading

● M. O. Goerbig, QuantumQuantum HallHall EffectsEffects, Les Houches Summer School (2009), chapterschapters 1-31-3 http://arxiv.org/abs/0909.1998 ● S. M. Girvin, TheThe QuantumQuantum HallHall Effect:Effect: NovelNovel ExcitationsExcitations andand BrokenBroken SymmetriesSymmetries, Les Houches Summer School (1998) http://arxiv.org/abs/cond-mat/9907002

● G. Murthy and R. Shankar, Rev. Mod. Phys. 75, 1101 (2003) http://arxiv.org/abs/cond-mat/0205326 ● D. Yoshioka, TheThe QuantumQuantum HallHall EffectEffect, Springer, Berlin (2002)

Ways to understand the QHE

Circular trajectories of electrons in a magnetic Field (cyclotron motion)

→ Bohr-Sommerfeld quantization (see D. Basko’s class)

(single-valuedness of wave function)

Landau quantization (“quick and dirty” approach)

Circular trajectories of electrons in a magnetic Field (cyclotron motion)

→ Bohr-Sommerfeld quantization (see D. Basko’s class)

(single-valuedness of wave function)

→ Quantization of cyclotron radius:

weird stuff integer integer (related to Berry phase)

magnetic length Landau quantization (“quick and dirty” approach)

Circular trajectories of electrons in a magnetic Field (cyclotron motion)

→ Bohr-Sommerfeld quantization (see D. Basko’s class)

(single-valuedness of wave function)

→ Quantization of cyclotron radius:

→ Quantization of energy: weird stuff (related to Berry phase)

here: Landau level quantization (canonical)