The quantum Hall effect (QHE)

Transcription

1 24/10/14 The quantum Hall effect (QHE) Presentation created for the Theoretisches-Physikalisches Seminar on problems in quantum mechanics Daniel Issing 1

2 Overview 1. Introduction Classical Hall effect & quantum mechanical changes 2. Creating a 2-dimensional electron gas (2DEG) 3. Properties of the 2DEG: Classical & qm description 4. The integer quantum Hall effect( IQHE) Explaining the existence of plateaus 5. A look ahead: The fractional QHE 6. Summary 2

3 Introduction Classical Hall Effect 3

4 Hall resistance Introduction However (von Klitzing et al., 1980), for high magnetic fields and low temperatures, one observes the following: Carrier Concentratio n n_1 Magnetic field B 4

5 Introduction 5

6 Fabrication of the 2DEG Several possibilities: MOSFETs, Heterojunctions, surface of liquid helium,... Here: Only MOSFET (Metal Oxid Semiconductor Field Effect Transitor) 6

7 Fabrication of the 2DEG 7

8 Fabrication of the 2DEG The Fermi-Dirac distribution 8

9 Fabrication of the 2DEG Band structure of a MOSFET 9

10 Beschreibung des 2DEG Classical Analysis. Homogenous B-Field Equation of motion: Solution: Lagrange function Hamilton operator Angular momentum (z-component) for symmetric gauge Cyclotron rotation 10

11 Description of the 2DEG Additional electrical field (homogenous) EoM Solution Drift velocity current density -> Classical Hall effect! 11

12 Description of the 2DEG Quantum mechanical analysis (magnetic field only) Hamilton operator Dynamic momentum Larmor length Pseudo momentum : Landau levels 12

13 Description of the 2DEG 13

14 Description of the 2DEG Additional electric field with Landau gauge Energy eigenvalues Expectation value for the velocity Entries of the resistance tensor with Klitzing constant Quantized resistance at the Hall plateaus 14

15 Integer QHE Separation (splitting) of the continuous density of states into Landau levels High field Density of states without external magnetic field (in 2D): 15

16 Integer QHE Landau fan Localized and extended states 16

17 Integer QHE 17

18 Fractionaler QHE Plateaus also exists for fractions of integers! -> Interactions between different electrons can no longer be ignored 2 approaches: wave function ansatz (Laughlin 1982) Composite Particles 18

19 Sources for the images [1] [3] ( ) [5,18] ( ) [5] ( ) [6] ( ) [7] ( ) [8] Verteilung_(Temperatur).svg.png ( ) [9] D. Yokoshika, The Quantum Hall Effekt, Springer 2004 [10,11,13] ( ) [15] ( ) [16,4] ( ) [17] MarkO. Goerbig, Quantum Hall Effekt. [cond-mat] v2 The number in square brackets refers to the page where the image appeared 19

20 References K.v. Klitzing, G. Dorda, M. Pepper, Physical Review Letter, Volume 45, Page 494 D.Yoshioka: The Quantum Hall Effekt, Springer 2004 K.v. Klitzing, Der Quanten Hall Effekt, Spektrum der Wissenschaft 1986, S.46 Anleitung_Quanten-Hall-Effekt.pdf ( ) Anleitung_Quanten-Hall-Effekt.pdf ( ) 20