lecture 2 2d electrons in excited landau levelscmp2008/lecturenotes/eisenstein_lecture 2.pdf ·...

Lecture 2

2D Electrons in Excited Landau Levels

What is the Ground State of an Electron Gas?

Wigner

lower density

Two Dimensional Electrons at High Magnetic Fields

Hartree-Fock prediction: Charge density waves throughout lowest Landau level

Fukuyama, Platzman, and Anderson, 1979

N=0

N=1

N=2

E Landau levels

Reality: Fractional Quantum Hall Liquids

lowest Landau level

Hartree-Fock Spectacularly Wrong!

Low Field Regime

Excited Landau levels

lowest Landau level

N=0

N=1

N=2

E Landau levels

800

600

400

200

0Long

itudi

nal R

esis

tanc

e (O

hm)

5.04.54.0Magnetic Field (Tesla)

Even-Denominator FQHE in N=1 LL

ν=5/2 20mK

300

200

100

0

Rxx

(Ω

)

1086420B (Tesla)

N=0N=123...

5/2 ν=2

5/3 4/3

T=150mKN=2 Landau level

11/2 9/2

Higher Landau Levels

Mobility ~ 10 7 cm2/Vs

Structure in Rxx in N ≥ 2 Landau levels → correlations

Structure in High Landau Levels

• Rxx at half filling increases dramatically below 100mK.

• Complex structure surrounds the peak.

• Peak width does not approach zero as T→ 0.

• Not consistent with the localization transition between IQHE states.

2.2 2.4 2.6 2.80

500

1000

B (Tesla)

25 mK50 mK80 mK

100 mK

ν=5 ν=4

Rxx (Ω)

ν = 9/2

400

200

0

2.62.4

T = 200 mK

2.62.4 2.62.4 2.62.4

T = 100 mK T = 80 mK T = 25 mK

Magnetic Field (Tesla)

Res

ista

nce

(Ohm

s)

200

0

<110>

<110>BAnisotropy

ν = 9/2

Rapid Onset Below 100mK

1000

800

600

400

200

0

Long

itudi

nal R

esis

tanc

es

(Ω)

2001000Temperature (mK)

<110>

<110>

ν = 9/2

Low temperature resistance anisotropy is consistently oriented relative to GaAs crystal axes.

Anisotropy Widespread in High Landau Levels

ν = 4 is a boundary between different transport regimes.

N = 0 & 1N = 2, 3, ...


Rxx

& R

yy(O

hms)

ν = 9/2

7/2

11/2

5/2

13/2

ν=4

T=25mK

<110>

<110>B1200

1000

800

600

400

200

0543210

1000

500

0

resis

tanc

es (

Ω)

543210magnetic field (Tesla)

200

150

100

50

0

resis

tanc

es (

Ω)

2.82.62.42.22.0magnetic field (Tesla)

More Unusual Features in High Landau levels

• Isotropy in flanks of LL

• New FQHE states?

Re-entrant Integer Hall Quantization


Integer QHE,localized electrons


no QHE,delocalized electrons


Integer QHE,re-localized electrons


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

Nodes in high LL wavefunctions soften short range Coulomb repulsion between electrons. Exchange energy favors phase separation.

N=5 LL

Stripes to Bubbles to Wigner Crystal

4 5 4 5 44 5 4 5 4 5 4

ν = 4+½

4+ε

Numerical Simulation: N=10 Landau Level

νN = 1/2 stripes

Koulakov, Fogler, Shklovskii

νN = 1/4 bubbles

νN = 1/16 Wigner crystal

400

200

0

Long

itudi

nal r

esis

tanc

e(Ω

)

2.52.01.5Magnetic field (Tesla)

Hall resistance (h/e

2)

1/4

1/5

1/61/7

9/2

11/213/215/2

N=3 N=2

State of the Art Samples

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?

1000

800

600

400

200

0

long

itudi

nal r

esis

tanc

es (Ω

)

2000

Sample A

Temperature (mK)1000

Sample B

Consistent Orientation of Anisotropy

<110>

<110>B

Independent of weak, high temperature anisotropies

Sample A

What about Surface Morphology?

Sample B

<110>

<110>

20μm

Surfaces are typically rough,Δz ~ 10 nm. 2DEG is buried.

Roughness is not isotropic.MBE growth kinetics is anisotropic,wafers can be miscut,etc.

Sample A

What about Surface Morphology?

Sample B

<110>

<110>

20μm

Sample A

What about Surface Morphology?1000

800

600

400

200

0Long

itudi

nal R

esis

tanc

es (

Ohm

s)

2.52.01.5Magnetic Field (Telsa)

5-10-93-2

3000

2500

2000

1500

1000

500

0Long

itudi

nal R

esis

tanc

es (O

hms)

2.52.01.5Magnetic Field (Tesla)

9-20-99-1 QW

Sample B

<110>

<110>

20μm

No systematic correlation.

Crystal Symmetry

GaAs has a zinc-blende crystal structure.

S4 symmetry ensures that band structure ε(k) is 4-fold symmetric.

Ga

As

+

AlGaAs GaAs

AlGaAsGaAsAlGaAs

+ +

Symmetry of Confinement Potential?

Does this eliminate all pinning mechanisms based upon lack of inversion symmetry in conventional heterointerfaces?

Kroemer, 19991000

03.53.02.5

ν = 9/2

7/2

[110][110]

1000

03.53.02.5

ν = 9/2

7/2

[110][110]

In-plane Magnetic Fields Can Switch “Hard” and “Easy”Transport Directions

1000

500

02.52.4 2.52.4 2.52.4

B||=0 B||=0.5T B||=1.7Tν = 9/2

<110>

<110>

Long

itudi

nal R

esis

tanc

e (

Ohm

s)

Perpendicular Magnetic Field (Tesla)

B||

B⊥

<110>

<110>

High resistance direction along B||

1000

500

0

600

400

200

0

300

200

100

0

300

200

100

043210

B|| (Tesla)

Long

itudi

nal R

esis

tanc

e (O

hms)

<110>

<110>

9/2

15/2

13/2

11/2

Same Effect in Many High Landau Levels

B|| along <110>

Theory of In-Plane Magnetic Field Effect

<110>

<110>

φ

cos(2 )φ= −E A

native symmetry breaker


B||

<110>

<110>

φλ

cos(2 ) cos(2 )φ λ= − +E A C

native symmetry breaker field anisotropy energy


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|| ~ 0.5 T

B|| along <110> B|| along <110>

300

200

100

043210 43210

B|| (Tesla) B|| (Tesla)

15/2

600

400

200

0

11/2

Long

itudi

nal R

esis

tanc

e (O

hms)

<110>

<110>

Dramatic Sensitivity to Direction of B||

1000

500

0

300

200

100

0

B|| (Tesla) B|| (Tesla)

9/2

13/2

Long

itudi

nal R

esis

tanc

e (O

hms)

0 1 432 0 1 432

It’s not that simple...

<110>

<110>

B|| along <110> B|| along <110>

6420B || (T) along [110]

6 4 2 0B || (T) along [110]

1200

400

R (Ω

) Rxx Ryy

Lower Density Sample

ν = 9/2

Low Density High Density

Density-Dependent Interchange of Anisotropy Axes

hard axis <110>hard axis <110>

Zhu, et al. 2002

Piezoelectricity of GaAs

Rashba and Sherman ‘87

Fil ‘00

[010]

[100]

θ

<110>

<110>

<100>

π/4−π/4 0

UPE

<110><110>

Piezoelectricity of GaAs

Rashba and Sherman ‘87

Fil ‘00

[010]

[100]

θ

<110>

<110>

<100>

π/4−π/4 0

UPE

<110><110>But what lifts the degeneracy?

<110><110>

Metastability in Double Well Potential<110>

<110>

800

600

400

200

0

R (Ω

)

2.01.91.8B (T)

θ = 7o

13/2

down sweep

800

600

400

200

0

R (Ω

)

2.01.91.8B (T)

θ = 7o

13/2

up sweepUp Sweep Down Sweep

1500

1000

500

0

R (Ω

)

2.01.91.8B (T)

θ = 7o

field cooled

13/2

Field Cooled

<110>

<110>

800

600

400

200

0

R (Ω

)

151050Time (hours)

B⊥ = 1.90 TT = 50mK

800

600

400

200

0

R (Ω

)

2.01.91.8B (T)

θ = 7o

13/2

down sweep

1500

1000

500

0

R (Ω

)

2.01.91.8B (T)

θ = 7o

field cooled

13/2

Very Slow Approaches to Equilibrium

Are the Anisotropic States Nematic Liquid Crystals?

isotropic

smectic

nematic

crystal

Liquid Crystal Phases of 2D ElectronsFradkin and Kivelson

A Nematic to Isotropic Phase Transition?

1000

800

600

400

200

0

Long

itudi

nal R

esis

tanc

es

(Ω)


ν=9/2

<110><110>

Hartree-Fock estimates of stripe formation temperature ≈ 2K.

Wexler and Dorsey

1000

800

600

400

200

0

Res

ista

nces

(Ω

)

500400300200100

Temperature (K)

Parallel Field Extends Anisotropy to Higher Temperatures

B|| = 0

B|| = 4.3T

ν = 9/2

M

TTc

Ferromagnet in an External Magnetic Field

B>0B=0

Comparison with classical 2DXY model

U = J ∑ cos(2[θi-θj]) - h ∑ cos(2θi)< i j> i

Similarity suggests that microscopic stripe moments exist at high temperatures.

6003000T (mK)

1050T/J

0.5

0.0

1.0

ρ hard

(kΩ

/)

M = <cos(2θ)>

0.050.5

1.5

h/J = 4.5

1.0

0.5

0.0

B|| = 0.0 1.3 2.2 4.3 5.4 9.2 T

Scaling



Rha

rd

(kΩ

)

3001500T/B|| (mK/T)

3001500T/B|| (mK/T)

Rha

rd

(kΩ

)

0.5

0.5

0

0

B|| along <110> B|| along <110>

Nature of Insulating Phases

Non-linear I-V Characteristics in Re-Entrant IQHE

Abrupt, hysteretic, and noisy transitions between insulating and conducting states.

100

50

0

Vyy

(μV

)6004002000

current (nA)

ν ≈ 4+1/4T = 25mK200

0

Rxx

(Ω)

2.92.82.72.62.5B (T)

ν = 9/2RIQHE RIQHE

Iy

Ex

Vyy

CDW Depinning due to Hall Electric Field?

10005000040

2.15

2.10

2.05

2.00

ν = 4

RIQHE

50 μV

Mag

netic

Fie

ld (T

)

Ryy (Ohms) Idc (nA)

Vyy

2.10

2.05

2.15

2.00

Mag

netic

Fie

ld (T

)

Vyy

0 10.5Idc (μA)Ryy (Ω)

040

RIQHE

IQHE

Discontinuous I-V curves found only within the RIQHE

Vyy

10008006004002000

Idc (nA)

25 mK

45 mK

55 mK

60 mK

65 mK

20 μV0

And only at very low temperatures

Vac

1086420milliseconds

50μV

200

0

Rxx

(Ω)

2.92.82.72.62.5B (T)

ν = 9/2RIQHE RIQHE

Vdc

(μV

)

1.51.00.50.0

200

0

B = 2.83 TVdc

Vac

Idc

Idc (μA)

Narrow Band Noise in Insulating Phases

Vdc

(mV

)ƒ

(kH

z)

0

00 21 3 4

0.3

20

10

30

Idc (μA)

Noi

se in

Vac

ƒ (kHz)0 10 30 40

100 nV/(Hz)1/2 0.63μA0.84μA0.99μA

20

Spectral Analysis

8

6

4

2

150100500T (mK)

rms

nois

e (μ

V)

Noise confined to RIQHE

250

03.02.92.82.72.62.52.4

B (T)

8

6

4

2Res

ista

nce

(Ω)

rms noise (μV

)ν = 9/2

Origin of Noise

Washboard noise?ƒwb ~ J • a / e ~ 10 MHz for 20nA/mm

a

ƒexp ~ 10 kHz

Avalanche heating?Why restricted to RIQHE at mK temperatures?

Circuit oscillations?

Frequencies generally increase with current, but

Iceberg or droplet noise?Do low frequencies point to 100μm-size objects?

Insensitive to external circuit. Local variations observed.

New Physics in the N=1 Landau Level

1000

500

0

Long

itudi

nal R

esis

tanc

e (Ω

)

6543210Magnetic Field (T)

N = 0N > 1 N = 1

FQHECDW

First Excited Landau Level

800

600

400

200

0

5.04.54.0

T=20, 40, 100mK

5/2

7/3

Magnetic Field (T)

Long

itudi

nal R

esis

tanc

e (O

hms)

Robust 5/2 States

,..,

..( ) .i ji n

3z z− ↑ ↑∏Ψ ~

A Role for Spin?

Ψ ~,..,

..( ) .i ji n

2z z− ↑ ↑∏

1/3 state

5/2 state?

,..,

..( ) .i ji n

3z z− ↑ ↑∏Ψ ~

A Role for Spin?

Ψ ~,..,

..( ) .i ji n

2z z− ↑ ↑∏

1/3 state

5/2 state?

but perhaps spins are not polarized...

Haldane-Rezayi “Hollow Core Model” 1988

θ BTOT

1988: Tilt Sample to Increase Zeeman Energy

Tilting destroys 5/2 state? Ground state contains

reversed spins.

1/2 state: Composite Fermion Fermi Liquid - No QHE

5/2 state: BCS-like p-wave paired CF Liquid - QHE

Jain

Halperin-Lee-Read

Moore-Read

Rezayi-Haldane

Both are spin polarized states.

Today’s Theory

600

400

200

05.04.54.03.53.0

B⊥ (Tesla)

Res

ista

nce

(Ω)

1999: Revisit 5/2 State in Tilted Fields

Tilting destroys 5/2 and 7/2 FQHE states and produces strongly anisotropic transport

B||=0 B||=7.7T along <110>

<110> <110>2000

1500

1000

500

05.04.54.03.53.0

B⊥ (Tesla)

Rezayi and Haldane: Stripe state close in energy to paired FQHE state at B|| = 0.

0.35

0.30

0.25

Rxy

(

h/e2 )

4.03.83.63.43.2


h/4e2

h/3e2

ν = 7/23+1/5

3+4/5

50mK 15mK

Re-Entrant Integer Hall Quantization

Integer Hall quantization near

ν – 3 ≈ 2/7, 3/7, 4/7, and 5/7

New Collective Insulators:

Bubbles?

Wigner Crystals?

Δ5/2 ~ 300mK

Similar near 5/2

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.

lecture 2 2d electrons in excited landau levelscmp2008/lecturenotes/eisenstein_lecture 2.pdf ·...

Documents