Lecture 4

Spin Configuration of 2D Electrons at High Magnetic Field

ωc ≈ 15 K @ 10 T

Why is spin interesting at high B?

g μBB

(g – 2)μB B ≈ 15 mK

Free electrons

ωc = 200 K @ 10 Tesla

Why is spin interesting at high B?

GaAs electrons

ωc = 200 K @ 10 Teslag μBB = 3 K

Why is spin interesting at high B?

GaAs electrons

ωc = 200 K @ 10 Teslag μBB = 3 K

2eε

≈ 150 K

GaAs electrons

Coulomb interactions can produce non-trivial spin states – Halperin 1983

Why is spin interesting at high B?

Example: FQHE at ν = 8/5

ν = 8/5

Example: FQHE at ν = 8/5

θ Btot

A Cautionary Tale: Even-Denominator FQHE

5/2N=1

N=0

1987

Evidence for Reversed Spins

Tilting destroys 5/2 state? Ground state contains

reversed spins.

Haldane-Rezayi “Hollow Core Model” 1988

Anisotropic Phases in High Landau Levels

ν = 4 is a boundary between different transport regimes.

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

Magnetic Field (Tesla)

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

Long

itudi

nal R

esis

tanc

e (Ω

)

6543210Magnetic Field (T)

N = 0N > 1 N = 1

FQHECDW

Betwixt and Between

600

400

200

05.04.54.03.53.0

B⊥ (Tesla)

Res

ista

nce

(Ω)

Tilt-Induced Anisotropy at 5/2 and 7/2

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 a

spin-polarized paired ν = 5/2 FQHE state.

Plan: Use NMR to modify Zeeman Energy

RF to coil

Au wires to ohmics

Bext

BRF

Hyperfine Coupling

-5

-4

-3

-2

-1

0

BN (T

)

0.0012 4 6 8

0.012 4 6 8

0.12 4 6 8

1Temperature (K)

total 75As 71Ga 69Ga

B = 10 TH1 = A(I·S)δ(R)

EZ = gμB (B + BN)

δξN δBN δRxxδEZ

Spin Transition in the Half-Filled Landau Level

B ~ 10T

H1 ~ 1 μT

10

5

0

RX

X (k

Ω)

1050B (T)

T = 50 mKn = 1.3 x 1011 cm-2

ν = 1/2

Measure xx

ZEρ∂

∂ versus density at ν = 1/2

Composite Fermions

Chern-Simons singular gauge transformation:

Attach an even number of fictitious flux quanta to

each electron

B* = B - 2φ0n1 1 2CFν ν

= −

1, 2, 3,CF jν = = … 1 2 3, , ,2 1 3 5 7

jj

ν = =+

…

Jain, others

Half-filled Landau Level

CFν = ∞12

ν = B* = 0

Fermi sea of CFs

02F Fk k=

Halperin, Lee, Read, others

eε

= ∝∼2 2 2

*2F

FCF

kE Bm

∝*CFm B

Variable Density Sample

10

5

0

ρxx

(kΩ

/)

1050B (T)

ν = 1/2

No evidence of any critical points

6

4

2

0

Δρxx

(Ω/

)

3210Time (1000 sec.)

8

6

4

2

0

Δρxx

(Ω/

)

200-20Δf (kHz)

f0 = 29.307 MHz

1.0

0.5

0

Δρxx

/ρxx

(x1

03 )

200150100500Temperature (mK)

RDNMR Observables at ν = 1/2

At B = 4 T, ν = 1/2 state isnot fully spin polarized.

6

4

2

0

Δρxx

(Ω/

)

3210Time (1000 sec.)

RF on

RF off

τω

=+

12

2 11riseR

TT T 1fall Tτ =

1riseNfallN

τδξτξ = −

Transients

8

6

4

2

0

Δρxx

(Ω/

)

200-20Δf (kHz)

f0 = 29.307 MHz

Lineshape

75As

Quadrupole SplittingKnight Shift

Wavefunction Tomography

1.0

0.5

0

Δρxx

/ρxx

(x1

03 )

200150100500Temperature (mK)

Temperature Dependence

( )N Tξ ( )xx

ZTE

ρ∂∂

and

Collapse of ν = 1/2 RDNMR Signal at High Fields

1 xx

xx Z

SEρ

ρ∂

≡∂

45 mK100 mK

0.15

0.10

0.05

0

S (

K-1

)

12108642B (T)

Rapid Increase of T1 at High Field

2000

1500

1000

500

0

T 1 (

s)

12108642B (T)

45 mK100 mK

E

k

EF

EZ

E

k

E FE Z

Simple Model of Composite Fermion Spin Transition at ν = 1/2

increase density

2

*F

FCF

kE Bm

∼ ∼ZE B∼

*

0

1 2.3CFmm g

≥ ≈ν = ½ state spin polarized if

RDNMR Reveals Transition to Full Spin Polarization at ν = 1/2

1 xx

xx Z

SEρ

ρ∂

≡∂

45 mK100 mK

0.15

0.10

0.05

0

S (

K-1

)

12108642B (T)

Rapid Increase of T1 in Spin Polarized Phase

2000

1500

1000

500

0

T 1 (

s)

12108642B (T)

45 mK100 mK

+ − − +• = + +z zI S I S I S I S

Rapid Increase of T1 in Spin Polarized Phase

2000

1500

1000

500

0

T 1 (

s)

12108642B (T)

45 mK100 mK

+ − − +• = + +z zI S I S I S I S

6

5

4

3

2

1

0

T 1-1

(1

0-3 s

-1)

200150100500T (mK)

3 T 4 T 5 T 6.4 T

Korringa-like Nuclear Spin Relaxation

11T aT b− = + in partially polarized phase

Unresolved Issues

• Sign of RDNMR signal

• Origin of peak in the RDNMR signal near the transition

• Density independence of T1 in partially polarized phase

Spin Polarization of 2D Electrons sans Landau Quantization

B||

Das Sarma and Hwang – PRB 2005

Screened charged impurity scattering reveals polarization transition

||

0xx xx

ZB Eρ ρ∂ ∂

= ≥∂ ∂

Phase Separation near Critical Point?

N=NcξCF = 1

N>NcξCF < 1

N<NcξCF = 1

δN

x

ξ

1

Spin and the Bilayer Excitonic Transition at νtot = 1

layer spacing0Quantum critical point

νT = 1/2 + 1/2νT = 1

RF

heater

What About Real Spin?

S

A

S

A

EZeeman

ΔSAS S

S

A

A

ΔSAS

EZeemanν = 1

In present samples:

ΔSAS < 100 μK Ezeeman ~ 1 K

Resistively Detected NMR

d/l = 1.92 +0.33 kHz/s730

720

710

700

690

RX

X (

)

40.0039.9839.9639.9439.92frequency (MHz)

71GaR

xx(O

hm)

40

20

0

R XX (k

)

3.23.02.8B (T)

νT = 1

Rxx

(kO

hm)

H1 = A(I·S)δ(R)

EZ = gμB (B + BN)

-5 0 5

NT = 5.4NT =10.9 x 10 10cm-2

NT = 6.9

-5 0 5

NT = 6.4

Crossing the νT =1 Phase Boundary

T=40mKTu

nnel

ing

Con

duct

ance

(10-

7 Ω

-1)

Interlayer Voltage (mV)

12

8

4

0

G0

(10-6

Ω−1

)

Heat onHeat off

12

8

4

0

G0

(10-6

Ω−1

)

6420Time (10

3s)

71Ga NMR pulses

Tunneling Experiment: RF and Heat Pulses

d/l = 1.92, T = 35 mK

V

dI/d

V

Moving the Phase Boundary

d/l = 1.98, T = 35 mK

60

40

20

0

dI/d

V (1

0-9 Ω

−1)

-0.2 -0.1 0.0 0.1 0.2Interlayer Bias (mV)

Equilibrium 60

40

20

0dI

/dV

(10-9

Ω−1

)-0.2 -0.1 0.0 0.1 0.2

Interlayer Bias (mV)

Hot Nuclei

Depolarizing nuclei temporarily produces excitonic phase.

Phase Boundary Depends on Zeeman Energy

Increasing the Zeeman energy shifts critical point to larger d/l

10-8

10-7

10-6

10-5

G0 (

10-9

Ω−1

)

2.001.981.961.941.921.90

d/l

T = 35 mK

Equilibrium Heat NMR

Simple Model

Competition between two phases with different electronic spin polarization

NMR or heat pulse increases fraction of sample in excitonicphase

Excitonic phase has the larger spin polarization

Compressible phase is not fully polarized.

2.12.01.9d/

2000

1000

0

T 1

(sec

.)

1.11.00.9νT

-1

3000

2000

1000

0

Rxx

(Ω

)

3.23.02.8Magnetic Field (T)

-0.5

0.0

ΔRxx /R

xxνT = 1

Nuclear Spin-Lattice Relaxation Rate

fast relaxation in compressible phase

peak reminiscent of skyrmion physics in single layer systems

Spin Polarization Transition at ν = ½in a Single 2D Layer

0.6

0.4

0.2

0.0

(ΔR

XX/

RX

X)/B

(arb

. uni

ts)

12108642B (T)

2000

1500

1000

500

0

T1 (s)

I. V. Kukushkin et al.; S. Melinte et al.; N. Freytag et al.

d/l

ΔSAS

QHE

NO QHE

Phase Diagram at νT = 1

d/l

ΔZ

ΔSAS

d/l

ΔSAS

QHE

NO QHE

Phase Diagram at νT = 1

d/l

NO QHE

spin polarized CFsCF paramagnet

spin polarized,pseudo-ferromagnet(excitonic superfluid)

1 Δ ΔZ SAS

ferromagnet QHE

Phase Diagram at νT = 1

Conclusions

• RDNMR an effective probe of spin physics in 2D electron systems

• ν = 1/2 is not spin polarized for B < 6T

• Excitonic Phase Transition in Bilayers at ν = 1 is spin dependent.