2D Electron Systems: Magneto-Transport Quantum Hall Effects

Transcription

1 Hauptseminar: Advanced Physics of Nanosystems 2D Electron Systems: Magneto-Transport Quantum Hall Effects Steffen Sedlak

2 The Hall Effect P.Y. Yu,, M.Cardona, Fundamentals of Semiconductors, Springer Verlag, Berlin, 1996 p.234 2

3 Performing a Quantum Hall Experiment Main ingredients: two-dimensional electron system low temperature high magnetic field 3

4 von Klitzing's experiment set up K. von Klitzing: 25 Years of Quantum Hall Effect (QHE), Séminaire Poincaré 2 (2004) 1 16 Si-MOSFET used as Hall device 4

5 von Klitzing's experiment set up P.Y. Yu,, M.Cardona, Fundamentals of Semiconductors, Springer Verlag, Berlin, 1996 p. 539 Sketch of the sample; topview 5

6 von Klitzing's experiment set up band structure formation of a 2DEG P.Y. Yu,, M.Cardona, Fundamentals of Semiconductors, Springer Verlag, Berlin, 1996 p. 539 Cut through the sample P.Y. Yu,, M.Cardona, Fundamentals of Semiconductors, Springer Verlag, Berlin, 1996 p

7 von Klitzing's experiment results P.Y. Yu,, M.Cardona, Fundamentals of Semiconductors, Springer Verlag, Berlin, 1996 p. 540 IQHE discovered in 1980 Nobel prize in

8 A similar experiment T. Ihn, Semiconductor Nanostructures, Oxford University Press, London, 2010, p.306 Hall measurement on a GaAs/AlGaAs heterostructure 8

9 IQHE - Phenomenon 2DEG, low temperature, high magnetic field changing B-field / gate voltage quantisation of Hall Resistance wide plateaus, spaced by von Klitzing constant or resistance quantum of use for metrology fixed to Ω 9

10 towards an explanation introduction of Landau levels consider disorder/localised states consider finite sample size 10

11 Broadening of Landau levels P.Y. Yu,, M.Cardona, Fundamentals of Semiconductors, Springer Verlag, Berlin, 1996 p

12 a simple simulation Modulus of wave function for states at different energies The following DOS follows from the simulation T. Ihn, Semiconductor Nanostructures, Oxford University Press, London, 2010, p. 303 T. Ihn, Semiconductor Nanostructures, Oxford University Press, London, 2010, p

13 finite sample size edge states T. Ihn, Semiconductor Nanostructures, Oxford University Press, London, 2010, p. 312 Edge states lead to perfectly conducting 1D-channels T. Ihn, Semiconductor Nanostructures, Oxford University Press, London, 2010, p

14 Going further fractional QHE pimp up the set up: higher magnetic field lower temperature high mobility sample ( >106cm²/Vs) 14

15 First publication of the FQHE discovered in 1981 by Dan Tsui and Horst Störmer theoretically described by Robert Laughlin in 1983 Nobel prize in 1998 H. L. Störmer: Nobel lecture: The fractional Quantum Hall effect, Rev. of Mod. Phys., vol. 71, no. 4,

16 fractional QHE - phenomenon more plateaus besides integer filling factors fractional filling factors T. Ihn, Semiconductor Nanostructures, Oxford University Press, London, 2010, p

17 general approach plateaus follow from energy gaps for ν<1: all electrons in lowest (spin-polarised) Landau level all electrons have the same energy e--e--interaction relevant for e--dynamics FQHE is a many-body problem (theoretically described by Bob Laughlin) 17

18 Concept of composite particles H. L. Störmer: Nobel lecture: The fractional Quantum Hall effect, Rev. of Mod. Phys., vol. 71, no. 4, 1999 Why you will need "mental flexibility... 18

19 Concept of composite particles H. L. Störmer: Nobel lecture: The fractional Quantum Hall effect, Rev. of Mod. Phys., vol. 71, no. 4, 1999 H. L. Störmer: Nobel lecture: The fractional Quantum Hall effect, Rev. of Mod. Phys., vol. 71, no. 4,

20 Concept of composite particles Putting together a new entity: the composite particle H. L. Störmer: Nobel lecture: The fractional Quantum Hall effect, Rev. of Mod. Phys., vol. 71, no. 4, 1999 H. L. Störmer: Nobel lecture: The fractional Quantum Hall effect, Rev. of Mod. Phys., vol. 71, no. 4,

21 - from e to CPs simple objects external field highly interacting complex objects no external field non-interacting H. Störmer says: 21

22 between bosons and fermions flux quantum extra "twist" H. L. Störmer: Nobel lecture: The fractional Quantum Hall effect, Rev. of Mod. Phys., vol. 71, no. 4,

23 between bosons and fermions T. Ihn, Semiconductor Nanostructures, Oxford University Press, London, 2010, p. 323 H. L. Störmer: Nobel lecture: The fractional Quantum Hall effect, Rev. of Mod. Phys., vol. 71, no. 4,

24 the FQHE state at ν=1/3 three vortices per electron composite bosons CBs condense into a new ground state characteristic energy gap FQHE state! H. L. Störmer: Nobel lecture: The fractional Quantum Hall effect, Rev. of Mod. Phys., vol. 71, no. 4,

25 even more vortices? H. L. Störmer: Nobel lecture: The fractional Quantum Hall effect, Rev. of Mod. Phys., vol. 71, no. 4, 1999 treat the remaining flux quanta as an external B-field acting on the CPs back to IQHE explains all ν = i +/- 1/q states 25

26 the FQHE state at ν=1/2 two vortices per electron composite fermions do not condense into a ground state (Pauli) field incorporated: equivalent to electrons at zero magnetic field! 26

27 (all) other states T. Ihn, Semiconductor Nanostructures, Oxford University Press, London, 2010, p. 323 CFs around ν=1/2 encounter a IQHE explains all ν = p/(2p +/- 1) states FQHE of electrons = IQHE of CFs 27

28 a peculiar FQHE state (ν=5/2) H. L. Störmer: Nobel lecture: The fractional Quantum Hall effect, Rev. of Mod. Phys., vol. 71, no. 4, 1999 may have similarities with Cooper pairs (superconductivity), but unclear 28

29 take-home-messages set up: 2DEG in high magnetic field at low temperature IQHE quantisation of Hall resistance stabilised through localised states quantisation depends only on e, h metrology electron-electron interaction many-body problem simplified picture: composite particles quantum effects on a macroscopic scale FQHE: T. Ihn, Semiconductor Nanostructures, Oxford University Press, London, 2010, p

30 sources R.B. Laughlin: Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations, Phys. Rev. Letters, vol. 50, no. 18, 1983 H. L. Störmer: Nobel lecture: The fractional Quantum Hall effect, Rev. of Mod. Phys., vol. 71, no. 4, 1999 K. von Klitzing: 25 Years of Quantum Hall Effect (QHE), Séminaire Poincaré 2 (2004) 1 16 D.C. Tsui et al.: Two-Dimensional Magnetotransport in the Extreme Quantum Limit, Phys. Rev. Letters, vol. 48, no. 22, 1982 T. Ihn: Semiconductor Nanostructures, Oxford University Press, New York, 2010 P.Y. Yu, M. Cardona: Fundamentals of Semiconductors, Springer Verlag, Berlin, 'QuantumHallEffectExplanationWithLandauLevels' 30