Beyond the Quantum Hall Effect

Transcription

1 Beyond the Quantum Hall Effect Jim Eisenstein California Institute of Technology School on Low Dimensional Nanoscopic Systems Harish-chandra Research Institute January February 2008

2 Outline of the Lectures I. Introduction to the Quantum Hall Regime a. Overview b. QHE Basics: Integer and Fractional, composite fermions. c. Thermal transport in quantum Hall edge channels d. Beyond the standard paradigm: Excited Landau Levels and Double Layers II. 2D Electrons in Excited Landau Levels a. Basic Observations. b. CDW scenario c. Symmetry Breakers d. Liquid crystal scenario e. Re-entrant insulating phases f. N = 1 Landau level III. Exciton Condensation in Bilayer Electron Systems a. Overview and history. Discovery of ν T = 1/2 and ν T = 1 QHE states. b. Quantum Hall ferromagnetism and the ν T = 1 QHE state. c. ν T = 1 QHE state as an exciton condensate. d. Tunneling e. Counterflow f. Finite temperature phase transition

3 Lecture 1 Introduction to the Quantum Hall Regime

4 Two-Dimensional Electron Gas

5 A Semiconductor Sandwich A B A CB ~100 Å VB

6 Quantum Confinement subband wave function CB ~100 Å VB

7 Quantum Confinement CB donor layers VB

8 Perfect Registry

9 Molecular Beam Epitaxy

10 Molecular Beam Epitaxy spray painting with atoms heated cells Ga Al As AlGaAs GaAs 100 A AlGaAs ultra high vacuum high quality GaAs substrate hot! allows for precision engineering of crystal layer by layer

11 Mobility of Electrons in GaAs μ = eτ * m μ = 36 x 10 6 cm 2 /Vs 10 nm mean free path ~ ¼ mm Loren Pfeiffer and Ken West

12 Hall Effect Measurements B V H V H I V xx V xx B R H = V H / I = B/ne

13 The Integer Quantum Hall Effect R xx (kω) R H (h/e 2 ) V H B I V xx 2 h/e RHall = = Ohms / integer j

14 2D Electrons in a Magnetic Field Circular orbits + de Broglie waves discrete radii and energies DOS Landau levels Energy Landau levels are massively degenerate: n 0 = eb/h. Precisely fill an integer number of LLs: n = j x n 0 R H = B/ne = B/jen 0 = h/je 2 IQHE associated with fully filled Landau levels.

15 Magnetic Fields Enhance Interactions DOS Fermi gas Energy DOS Landau levels Energy Magnetic fields quench the KE and produce a strongly interacting system

16 Fractional Quantum Hall Effect 1/3 N= 0 FQHE associated with partially filled Landau levels.

17 Laughlin s Wavefunction Ψ(z 1,z 2,,z n ) ~ i< j ( z z ) i j 3

18 Laughlin s Wavefunction A bizarre fluid state of many electrons having fractionally charged excitations.

19 Odd-Denominator Rule 1/2 No FQHE at half-filling of lowest Landau level.

20 Landau Levels B ( p+ ea) 2 1 H = + g μb σ Β + V( z) 2m εν = N + ωc ± g μbb+ E0 2 2 D= eb/ h ω μ c B = eb / m = e /2m 0 In GaAs: m * /m 0 = 0.067, g = N=2 N=1 N=0 gμ B B ω c ω c = B = 1T gμ B B ω c /70

21 Wavefunctions Landau gauge: A = yb xˆ ψ = e φ ( y y ) ζ ( z) χ ikx Nk,, σ N k φ ξ ξ ξ /2 N( ) = e HN( ) = Δ = 2 yk k yk = / eb 2π L x 2 σ Symmetric gauge: A = -½(r x B) 2 z /4 j ψ ( ) 0, j, = ze ζ0 zχ σ z = ( x+ iy)/ σ

22 Integer QHE and Edge States I = i = = = V 2 2 Ldy x k ε N e e e k 2 ( μr μl) H occ. 2π yk Lx h h

23 Disorder Determines Plateau Widths R xy B E x E F y y

24 Disorder Determines Plateau Widths R xy B E x E F y y

25 Disorder Determines Plateau Widths R xy B E x E F y y

26 Disorder Determines Plateau Widths R xy B E x E F y y

27 Laughlin s Wavefunction

28 Single Electron in Lowest Landau Level Symmetric gauge: A = -½(r x B) 2 j z /4 0,, σ = 0 ψ j z e ζ ( z) χ eb 0 j < nφ = π R h 2 σ

29 Single Electron in Lowest Landau Level ψ( z ) a z exp nφ 1 j z 1 j 1 j=

30 Filled Lowest Landau Level A unique Slater determinant: n ν = = z1 z2 z n Ψ ( z1, z2,, zn) = exp z z z n φ n 1 n 1 n n n j= 1 z 2 j 4 Ψ( z, z,, z ) ( z z ) 1 2 n i j i< j

31 Visualizing ν = 1 Ψ ( z, z,, z ) = ϕ( z ) χ( z,, z ) 1 2 n 1 2 n ϕ( z ) = ( z z )( z z ) ( z z ) n

32 Uncorrelated ν <1State n < n φ ϕ ( z ) ( z z )( z z ) ( z z ) P( z ) n 1 P(z 1 ) is an undetermined polynomial with n φ -n zeros

33 Laughlin correlated ν = 1/3 State P( z ) ( z z ) ( z z ) ( z z n ) Ψ( z, z,, z ) ( z z ) 1 2 n i j i< j 3

34 Excitations of the Laughlin Liquid Φ 0 Add one more flux quantum: w Ψ z1 z2 zn zk w zi zj k i< j (,,, ) ( ) ( ) A quasi-hole with charge q = +e/3 3

35 Excitations of the Laughlin Liquid B > B 1/3 quasiholes q = + e/3 B < B 1/3 quasielectrons q = - e/3 E tot q-h q-e A gap to charged excitations ν = 1/3 n φ Δ e ε

36 Many Fractional Quantum Hall States Laughlin states: ν = 1/m = 1/3, 1/5,... ν = 1-1/m = 2/3, 4/5,... Whence 2/5, 3/7 etc.?

37 Hierarchy Model Interactions amongst quasiparticles produce new condensates 1/3 2/7 2/5 3/11 5/17 5/13 3/7 Most hierarchy states are not observed

38 Composite Fermions Jain, others Chern-Simons singular gauge transformation: Attach an even number of fictitious flux quanta to each electron B * = B - 2φ 0 n 1 1 = ν CF ν 2 ν CF = j = 1, 2, 3, j ν = =,,, 2j

39 Half-filled Landau Level Halperin, Lee, Read, others 1 ν = ν CF = B 2 * = 0 Fermi sea of CFs k F = 2 k 0 F At ν = 1/2 quasiparticles move in straight lines. R c 1 ν 1/2

40 Semi-classical transport of CFs Dimensional resonances in an anti-dot lattice Kang, et al. 1993

41 High Landau Levels lowest Landau level

42 1987: Even-Denominator FQHE Willett, et al. 5/2 N=1 N=0

43 Transport Anisotropy in High Landau Levels T=25mK ν = 9/2 B <110> R xx & R yy (Ohms) /2 11/2 ν=4 <110> ν = 4 is a boundary between different transport regimes /2 5/ Magnetic Field (Tesla) N = 2, 3,... N = 0 & 1

44 Electronic Liquid Crystals Longitudinal Resistances (Ω) A nematic to isotropic transition? Temperature (mk) 200

45 Double Layer Two-Dimensional Electron Gas

46 QHE in Double Layer 2D Systems 2.5 1/ Hall Resistance (h/e 2 ) ν T = 1 x Diagonal Resistance (kω) ν T = 1 = ½ + ½ ν T = ½ = ¼ + ¼ Magnetic Field (Tesla)

47 Quantum Hall Superfluid Start with a double layer 2D electron gas Add a magnetic field A BCS-like superfluid comprised of interlayer excitons. + _ + _ + _ + _ + _ + _