Lecture 2 2D Electrons in Excited Landau Levels

Lecture 2 2D Electrons in Excited Landau Levels

What is the Ground State of an Electron Gas? lower density Wigner

Two Dimensional Electrons at High Magnetic Fields E Landau levels N=2 N=1 N= Hartree-Fock prediction: Charge density waves throughout lowest Landau level Fukuyama, Platzman, and Anderson, 1979

Hartree-Fock Spectacularly Wrong! lowest Landau level Reality: Fractional Quantum Hall Liquids

Low Field Regime lowest Landau level E Landau levels N=2 N=1 N= Excited Landau levels

Even-Denominator FQHE in N=1 LL Longitudinal Resistance (Ohm) 8 6 4 2 ν=5/2 2mK 4. 4.5 5. Magnetic Field (Tesla)

Higher Landau Levels Mobility ~ 1 7 cm 2 /Vs 3 N=2 Landau level 11/2 9/2 T=15mK 5/3 4/3 R xx (Ω) 2 1 5/2 ν=2... 3 2 2 N=1 4 6 B (Tesla) N= 8 1 Structure in R xx in N 2 Landau levels correlations

Structure in High Landau Levels 1 R xx (Ω) 5 ν = 9/2 25 mk 5 mk 8 mk 1 mk ν=5 ν=4 2.2 2.4 2.6 2.8 B (Tesla) R xx at half filling increases dramatically below 1mK. Complex structure surrounds the peak. Peak width does not approach zero as T. Not consistent with the localization transition between IQHE states.

Anisotropy B <11> <11> T = 2 mk T = 1 mk T = 8 mk T = 25 mk 4 Resistance (Ohms) 2 ν = 9/2 2 2.4 2.6 2.4 2.6 2.4 2.6 Magnetic Field (Tesla) 2.4 2.6

Rapid Onset Below 1mK Longitudinal Resistances (Ω) 1 8 6 4 2 ν =9/2 <11> <11> 1 Temperature (mk) 2 Low temperature resistance anisotropy is consistently oriented relative to GaAs crystal axes.

Anisotropy Widespread in High Landau Levels 12 1 T=25mK ν = 9/2 B <11> R xx & R yy (Ohms) 8 6 4 13/2 11/2 ν=4 <11> ν = 4 is a boundary between different transport regimes. 2 7/2 5/2 1 2 Magnetic Field (Tesla) 3 4 5 N = 2, 3,... N = & 1

More Unusual Features in High Landau levels 1 resistances (Ω) 5 1 2 3 4 magnetic field (Tesla) 5 2 resistances (Ω) 15 1 5 Isotropy in flanks of LL New FQHE states? 2. 2.2 2.4 magnetic field (Tesla) 2.6 2.8

Re-entrant Integer Hall Quantization

Re-entrant Integer Hall Quantization Integer QHE, localized electrons

Re-entrant Integer Hall Quantization no QHE, delocalized electrons

Re-entrant Integer Hall Quantization Integer QHE, re-localized electrons

Re-entrant Integer Hall Quantization Integer QHE, re-localized electrons RIQHE states must be collective insulators

Charge Density Waves in High Landau Levels Koulakov, Fogler, and Shklovskii; Moessner and Chalker 1996 N=5 LL Nodes in high LL wavefunctions soften short range Coulomb repulsion between electrons. Exchange energy favors phase separation.

Stripes to Bubbles to Wigner Crystal 4 5 4 5 4 5 4 4 5 4 5 4 ν = 4+½ 4+ε

Numerical Simulation: N=1 Landau Level ν N = 1/2 stripes ν N = 1/4 bubbles ν N = 1/16 Wigner crystal Koulakov, Fogler, Shklovskii

State of the Art Samples N=3 N=2 Longitudinal resistance (Ω) 4 2 15/2 1.5 13/2 11/2 2. 9/2 2.5 1/4 1/5 1/6 1/7 Hall resistance (h/e 2 ) Magnetic field (Tesla) A New Class of Collective Phases of 2D Electron Systems

Taking a Closer Look What Orients the Resistive Anisotropy? Are the Anisotropic States Nematic Liquid Crystals? Nature of Insulating Phases New Physics in the N=1 Landau Level

What Orients the Resistive Anisotropy?

Consistent Orientation of Anisotropy longitudinal resistances (Ω) 1 8 6 4 2 Sample A Sample B B <11> <11> 2 Temperature (mk) 1 Independent of weak, high temperature anisotropies

What about Surface Morphology? Sample A <11> Surfaces are typically rough, Δz ~ 1 nm. 2DEG is buried. Sample B <11> Roughness is not isotropic. MBE growth kinetics is anisotropic, wafers can be miscut, etc. 2μm

What about Surface Morphology? Sample A <11> <11> Sample B 2μm

What about Surface Morphology? Sample A <11> <11> Longitudinal Resistances (Ohms) 1 8 6 4 2 5-1-93-2 1.5 2. 2.5 Magnetic Field (Telsa) Sample B 2μm Longitudinal Resistances (Ohms) 3 25 2 15 1 5 9-2-99-1 QW 1.5 2. 2.5 Magnetic Field (Tesla) No systematic correlation.

Crystal Symmetry GaAs has a zinc-blende crystal structure. Ga As S 4 symmetry ensures that band structure ε(k) is 4-fold symmetric.

Symmetry of Confinement Potential? Kroemer, 1999 1 ν = 9/2 [11] + [11] AlGaAs GaAs 7/2 2.5 3. 3.5 1 ν = 9/2 [11] + + [11] AlGaAs GaAs AlGaAs 2.5 3. 7/2 3.5 Does this eliminate all pinning mechanisms based upon lack of inversion symmetry in conventional heterointerfaces?

In-plane Magnetic Fields Can Switch Hard and Easy Transport Directions <11> B <11> B Longitudinal Resistance (Ohms) 1 5 ν = 9/2 <11> B = B =.5T B =1.7T <11> 2.4 2.5 2.4 2.5 2.4 2.5 Perpendicular Magnetic Field (Tesla) High resistance direction along B

Same Effect in Many High Landau Levels 1 5 <11> <11> 9/2 B along <11> Longitudinal Resistance (Ohms) 6 4 2 3 2 1 11/2 13/2 3 2 15/2 1 1 2 3 4 B (Tesla)

Theory of In-Plane Magnetic Field Effect φ <11> <11> E = A cos(2 φ) native symmetry breaker

Theory of In-Plane Magnetic Field Effect B λ φ <11> <11> E = A cos(2 φ) + C cos(2 λ) native symmetry breaker field anisotropy energy

Theory of In-Plane Magnetic Field Effect Finite thickness of 2D electron layer allows B to distort circular cyclotron orbits: B Jungwirth, MacDonald and Girvin Stanescu and Phillips Shklovskii In agreement with experiment, theory predicts stripes prefer to be perpendicular to B. Estimated native anisotropy energy ~ 1mK/electron at B ~.5 T

Dramatic Sensitivity to Direction of B B along <11> B along <11> Longitudinal Resistance (Ohms) 6 4 2 3 2 1 <11> <11> 1 2 3 4 1 2 3 4 B (Tesla) B (Tesla) 11/2 15/2

It s not that simple... B along <11> B along <11> Longitudinal Resistance (Ohms) 1 5 3 2 1 <11> <11> 1 2 3 4 1 2 3 4 B (Tesla) B (Tesla) 9/2 13/2

Lower Density Sample ν = 9/2 12 R xx R yy R (Ω) 4 6 4 2 (T) along [11] B B 2 4 (T) along [11] 6

Density-Dependent Interchange of Anisotropy Axes Low Density High Density hard axis <11> hard axis <11> Zhu, et al. 22

Piezoelectricity of GaAs <11> Rashba and Sherman 87 Fil <1> <11> [1] <11> <11> U PE θ [1] π/4 π/4

Piezoelectricity of GaAs <11> Rashba and Sherman 87 Fil <1> <11> [1] <11> <11> U PE But what lifts the degeneracy? θ [1] π/4 π/4

Metastability in Double Well Potential <11> <11> 8 Up up Sweep sweep 13/2 θ = 7 o 8 Down down sweep Sweep θ = 7 o 13/2 15 Field Cooled field cooled 13/2 θ = 7 o R (Ω) 6 4 2 R (Ω) 6 4 2 R (Ω) 1 5 1.8 1.9 B (T) 2. 1.8 1.9 B (T) 2. 1.8 1.9 B (T) 2. <11> <11> <11> <11>

Very Slow Approaches to Equilibrium 8 down sweep 13/2 θ = 7 o 15 field cooled 13/2 θ = 7 o R (Ω) 6 4 2 R (Ω) 1 5 1.8 1.9 B (T) 2. 1.8 1.9 B (T) 2. 8 B = 1.9 T T = 5mK 6 R (Ω) 4 2 5 1 Time (hours) 15

Are the Anisotropic States Nematic Liquid Crystals?

Liquid Crystal Phases of 2D Electrons Fradkin and Kivelson crystal smectic nematic isotropic

A Nematic to Isotropic Phase Transition? Longitudinal Resistances (Ω) 1 8 6 4 2 ν=9/2 <11> <11> 1 Temperature (mk) 2 Hartree-Fock estimates of stripe formation temperature 2K. Wexler and Dorsey

Parallel Field Extends Anisotropy to Higher Temperatures ν=9/2 Resistances (Ω) 1 8 6 4 B = B = 4.3T M Ferromagnet in an External Magnetic Field 2 B= B> 1 2 3 4 5 T c T Temperature (K)

Comparison with classical 2DXY model U = J cos(2[θ i -θ j ]) - h cos(2θ i ) < i j> i ρ hard (kω/ ) 1..5 B =. 1.3 2.2 4.3 5.4 9.2 T 1.5.5 h/j = 4.5 1..5 M = <cos(2θ)>. 3 T (mk) 6.5 5 T/J. 1 Similarity suggests that microscopic stripe moments exist at high temperatures.

Scaling B along <11> B along <11> R hard (kω).5 3 6 Temperature (mk) 3 6 Temperature (mk) R hard (kω).5 15 3 T/B (mk/t) 15 3 T/B (mk/t)

Nature of Insulating Phases

Non-linear I-V Characteristics in Re-Entrant IQHE 1 ν 4+1/4 R xx ( Ω ) 2 RIQHE ν = 9/2 RIQHE V yy (μv) 5 T = 25mK 2.5 2.6 2.7 B (T) 2.8 2.9 2 4 current (na) 6 Abrupt, hysteretic, and noisy transitions between insulating and conducting states.

CDW Depinning due to Hall Electric Field? E x I y V yy

Discontinuous I-V curves found only within the RIQHE 2.15 ν = 4 IQHE Magnetic Field (T) Magnetic Field (T) 2.1 2.5 V yy yy RIQHE RIQHE 2. 4 4 R yy (Ohms) R yy (Ω) 5 μv 5 1.5 1 I dc (na) I dc (μa)

And only at very low temperatures 65 mk 6 mk V yy 55 mk 45 mk 25 mk 2 μv 2 4 6 8 1 I dc (na)

Narrow Band Noise in Insulating Phases R xx ( Ω) 2 RIQHE ν = 9/2 RIQHE 2.5 2.6 2.7 B (T) 2.8 2.9 2 I dc V dc B = 2.83 T V dc (μ V) V ac V ac 5μV..5 I dc 1. (μa) 1.5 2 4 6 milliseconds 8 1

Spectral Analysis Noise in V ac 1 nv/(hz) 1/2.63μA.84μA.99μA 1 2 ƒ (khz) 3 4 V dc (mv).3 3 ƒ (khz) 2 1 1 2 3 4 (μa) I dc

Noise confined to RIQHE Resistance (Ω) 25 2.4 2.5 2.6 ν = 9/2 2.7 B (T) 2.8 2.9 8 6 4 2 3. rms noise (μv) 8 rms noise (μv) 6 4 2 5 1 T (mk) 15

Origin of Noise a Washboard noise? ƒ wb ~ J a / e ~ 1 MHz for 2nA/mm Frequencies generally increase with current, but ƒ exp ~ 1 khz Iceberg or droplet noise? Do low frequencies point to 1μm-size objects? Avalanche heating? Why restricted to RIQHE at mk temperatures? Circuit oscillations? Insensitive to external circuit. Local variations observed.

New Physics in the N=1 Landau Level

First Excited Landau Level Longitudinal Resistance (Ω) 1 5 CDW N > 1 N = 1 FQHE N = 1 2 3 4 Magnetic Field (T) 5 6

Robust 5/2 States 8 T=2, 4, 1mK Longitudinal Resistance (Ohms) 6 4 2 5/2 7/3 4. 4.5 Magnetic Field (T) 5.

A Role for Spin? 1/3 state Ψ ~ i,.., n ( z z )... i j 3 Ψ ~ i,.., n ( z z )... i j 2 5/2 state?

A Role for Spin? 1/3 state Ψ ~ i,.., n ( z z )... i j 3 Ψ ~ i,.., n ( z z )... i j 2 5/2 state? but perhaps spins are not polarized... Haldane-Rezayi Hollow Core Model 1988

1988: Tilt Sample to Increase Zeeman Energy θ B TOT Tilting destroys 5/2 state? Ground state contains reversed spins.

Today s Theory 1/2 state: Composite Fermion Fermi Liquid - No QHE Jain Halperin-Lee-Read 5/2 state: BCS-like p-wave paired CF Liquid - QHE Moore-Read Rezayi-Haldane Both are spin polarized states.

1999: Revisit 5/2 State in Tilted Fields B = B =7.7T along <11> Resistance (Ω) 6 4 2 <11> <11> 2 15 1 5 3. 3.5 4. 4.5 B (Tesla) 5. 3. 3.5 4. 4.5 B (Tesla) 5. Tilting destroys 5/2 and 7/2 FQHE states and produces strongly anisotropic transport Rezayi and Haldane: Stripe state close in energy to paired FQHE state at B =.

Re-Entrant Integer Hall Quantization.35 5mK 15mK h/3e 2 Integer Hall quantization near ν 3 2/7, 3/7, 4/7, and 5/7 R xy (h/e 2 ).3 ν = 7/2 3+1/5 3+4/5 New Collective Insulators: Bubbles?.25 h/4e 2 Wigner Crystals? 3.2 3.4 3.6 3.8 4. Magnetic Field (Tesla)

Similar near 5/2 Δ 5/2 ~ 3mK

5/2 State Quasiparticles May Be Non-Abelian

Conclusion A new class of collective phases of 2D electron systems in high LLs. Impressive overlap with HF theory of stripe and bubble phases. N=1 LL is borderline; has FQHE and stripes and bubbles.