Introduction to Quantum Hall Effects 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 © The Big Bang Theory 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 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 ● 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 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) 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) 1.) Hamiltonian of electrons in a lattice: latticelattice momentummomentum Landau level quantization (canonical) 1.) Hamiltonian of electrons in a lattice: latticelattice momentummomentum → Example 1: electrons in the vicinity of a band minimum (e.g. GaAs) band mass Landau level quantization (canonical) 1.) Hamiltonian of electrons in a lattice: latticelattice momentummomentum → Example 1: electrons in the vicinity of a band minimum (e.g. GaAs) band mass → Example 2: electrons in graphene/2D boron nitride or transition-metal dichalcogenides (“gapped graphene”) E k y EF kx Landau level quantization (canonical) 1.) Hamiltonian of electrons in a lattice: latticelattice momentummomentum 2.) Presence of a magnetic field (→ Peierls substitution) with gauge-invariant not gauge-invariant Landau level quantization (canonical) 1.) Hamiltonian of electrons in a lattice: latticelattice momentummomentum 2.) Presence of a magnetic field (→ Peierls substitution) with 3.) Quantum mechanics: yields non-commuting components of Landau level quantization (canonical) Quantum mechanics of electrons in a magnetic field: and Landau level quantization (canonical) Quantum mechanics of electrons in a magnetic field: and Remarks: ● Result is gauge-invariant (i.e. depends only on not on ) ● and are conjugate q.m. variables → introduction of “harmonic-oscillator” ladder operators with Example 1: Landau levels Parabolic band: Equidistant energy (Landau) levels: “weird” offset 2 arises naturally from commutator Example 2: Landau levels (graphene) Landau levels of graphene: ~ band index 2 Example 2: Landau levels (graphene) Landau levels of graphene: “weird offset” 2 Example 2: Landau levels (gapped graphene) Landau levels of gapped graphene: Example 2: Landau levels (gapped graphene) Landau levels of gapped graphene: “Dirac mass” weird offset !!! Example 2: Landau levels (gapped graphene) Landau levels of gapped graphene: “parity anomaly”: does not disperse with magnetic field ! Unveiling Landau levels: magneto- optical spectroscopy (→graphene) see D. Basko’s class Landau-level degeneracy Origin: - translation invariance → energy does not depend on center of cyclotron motion Landau-level degeneracy Origin: - translation invariance → energy does not depend on center of cyclotron motion decomposition of position: : cyclotron coordinate (dynamics) Landau-level degeneracy Quantum mechanics: BUT: → Heisenberg uncertainty: Landau-level degeneracy → Heisenberg uncertainty: Consequences: ● Each quantum state occupies minimal surface ● Number of states per Landau level: total surface → number of flux quanta threading system Landau-level degeneracy → Heisenberg uncertainty: Consequences: ● Each quantum state occupies minimal surface ● Number of states per Landau level: ● Filling factor: Landau-level degeneracy → Heisenberg uncertainty: Consequences: ● Each quantum state occupies minimal surface ● Number of states per Landau level: ● Filling factor: ● Quantum states characterized by two orbital quantum numbers internal degrees of freedom (spin, valley,...) Landau-level quantum number guiding center (Landau site) 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 Quantum Hall electrons in electrostatic potential sample edges charged impurities → potential lifts Landau-level degeneracy → good approximation for Quantum Hall electrons in electrostatic potential sample edges charged impurities → potential lifts Landau-level degeneracy Heisenberg equations of motion: Quantum Hall electrons in electrostatic potential sample edges charged impurities → potential lifts Landau-level degeneracy Heisenberg equations of motion: → guiding centers follow equipotential lines (semiclassics) Quantum Hall electrons in electrostatic potential sample edges →→ extendedextended statesstates ~~ conductionconduction channelschannels charged impurities → guiding centers follow equipotential lines (semiclassics) Quantum Hall electrons in electrostatic potential sample edges →→ extendedextended statesstates ~~ conductionconduction channelschannels charged impurities →→ localizedlocalized statesstates ~~ electronselectrons trappedtrapped onon closedclosed equipotentialequipotential lineslines aroundaround potentialpotential valleys/hillsvalleys/hills → guiding centers follow equipotential lines (semiclassics) Edge states Four-terminal resistance measurement constance of chemical potential : no electron leakage through the bulk ! Four-terminal resistance measurement constance of chemical potential : no electron leakage through the bulk ! non-zero, but why quantized ? Conductance of a (single) Landau