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Nuclear Spin Ferromagnetic transition in a 2DEG

Pascal Simon LPMMC, Université Joseph Fourier

CNRS, Grenoble Department of Physics, University

of Basel

Collaborator Daniel Loss

GDR Physique Quantique Mésoscopique Aussois

21 Mars 2007

OUTLOOK

- I. THE HYPERFINE INTERACTION
- II. NUCLEAR SPIN FERROMAGNETIC PHASE TRANSITION
- IN A NON-INTERACTING 2D ELECTRON GAS ?
- III. INCORPORATING ELECTRON-ELECTRON INTERACTIONS
- IV. CONCLUSION

I. THE HYPERFINE INTERACTION

I. SPIN FILTERING

Central issue for quantum computing

decoherence of spin qubit

Sources of spin decay in GaAs quantum dots

- spin-orbit interaction (bulk structure)
- couples charge fluctuations with spin ?

spin-phonon interaction, but - this is weak in quantum dots (KhaetskiiNazarov,

PRB00) - and T22T1 (Golovach et al., PRL 93, 016601

(2004)) - contact hyperfine interaction important

decoherence source - (Burkard et al, PRB 99 Khaetskii et al., PRL

02/PRB 03 CoishLoss, PRB2004)

Hyperfine interaction for a single spin

Electron Zeeman energy

Hyperfine interaction

Nuclear spin dipole-dipole interaction

Separation of the Hyperfine Hamiltonian

Hamiltonian

Note nuclear field

is a quantum operator

Nuclear spins provide hyperfine field h with

quantum fluctuations seen by electron spin

Nuclear spins provide hyperfine field h with

quantum fluctuations seen by electron spin

Nuclear spins provide hyperfine field h with

quantum fluctuations seen by electron spin

With mean lthgt0 and quantum variance dh

Suppression due to a high magnetic field

- The hyperfine interaction is suppressed in the
- presence of a magnetic field
- (electron Zeeman splitting) since
- electron spin nuclear spin flip-flops do not
- conserve energy.

B

B

Total suppression requires full polarization of

nuclear spins which is not currently achievable

Polarization of nuclear spins

- 1. Dynamical polarization
- optical pumping lt65, Dobers et al. '88,

Salis et al. '01, Bracker et al. '04 - transport through dots 5-20, Ono Tarucha,

'04, Koppens et al., '06,... - projective measurements experiment?

2. Thermodynamic polarization i.e.

ferromagnetic phase transition? Q Is it

possible in a 2DEG? What is the Curie temperature?

Problem is quite old and was first studied in

1940 by Fröhlich Nabarro for bulk metals!

II. NUCLEAR SPIN FERROMAGNETIC PHASE TRANSITION

IN A NON-INTERACTING 2D ELECTRON GAS ?

I. SPIN FILTERING

A tight binding formulation

Kondo Lattice formulation

is the electron spin operator at site

RQ For a single electron in a strong confining

potential, we recover the previous description by

projecting the hyperfine Hamiltonian in the

electronic ground state

An alternative description for a numerical

approach ?

PS D.Loss, PRL 2007 (cond-mat/0611292)

A Kondo lattice description

This description corresponds to a Kondo lattice

problem at low electronic density

What is known ?

The ground state of the single electron case is

known exactly and corresponds to a ferromagnetic

spin state

Sigrist et al., PRL 67, 2211 (1991)

Several elaborated mean field theory have been

used to obtain the phase Diagram of the 3D Kondo

lattice

A ferromagnetic phase expected at small A/t and

low electronic density ?

Lacroix and Cyrot., PRB 20, 1969 (1979)

Effective nuclear spin Hamiltonian (RKKY)

Strategy A (hyperfine) is the smallest energy

scale We integrate out

electronic degrees of freedom

including e-e interactions (e.g. via a

Schrieffer-Wolff transformation)

Pure spin-spin Hamiltonian for nuclear spins only

'RKKY interaction'

(justified since nuclear spin dynamics is much

slower than electron dynamics)

Assuming no electronic polarization

An effective nuclear spin Hamiltonian

where

'RKKY interaction'

and

is the electronic longitudinal spin

susceptibility in the static limit (?0).

Free electrons Jr is standard RKKY interaction

Ruderman Kittel, 1954 Note that result is also

valid in the presence of electron-electron

interactions

2D What about the Mermin-Wagner theorem?

The Mermin-Wagner theorem states that there is no

finite temperature phase transition in 2D for a

Heisenberg model provided that

For non-interacting electrons, reduces to

the long range RKKY interaction

? nothing can be inferred from the Mermin-Wagner

theorem !

Nevertheless, due to the oscillatory character of

the RKKY interaction, one may expect some

extension of the Mermin-Wagner theorem,

and, indeed it was conjectured that in 2D Tc 0

(P. Bruno, PRL 87 ('01)).

The Weiss mean field theory.1

Consider a particular Nuclear spin at site

Mean field

Effective magnetic field

With

If we assume

One obtains a self-consistent mean field equation

The Weiss mean field theory.2

PS D Loss, PRL 2007

But is the simple MFT result really justified

for 2D ?

Spin wave calculations

The mean field calculations and other results on

the 3D Kondo lattice suggest a ferromagnetic

phase a low temperature. Let us analyze its

stability.

Energy of a magnon

The magnetization per site

Magnon occupation number

The Curie temperature is then defined by

Susceptibility of the non-interacting 2DEG

The 2D non-interacting electron gas

In the continuum limit

Electronic density in 2D

Expected and in agreement with the conjecture !

III.Incorporating electron-electron interactions

I. SPIN FILTERING

Perturbative calculation of the spin

susceptibility in a 2DEG

Consider screened Coulomb U and 2nd order pert.

theory in U

Chubukov, Maslov, PRB 68, 155113 (2003)

? give singular corrections to spin and

charge susceptibility due to

non-analyticity in polarization propagator

? (sharp Fermi surface) ? non-Fermi liquid

behavior in 2D

Correction to spin susceptibility in 2nd order

in U

Chubukov Maslov, PRB 68, 155113 (2003)

(remaining diagrams cancel or give vanishing

contributions)

Non-analyticities in the particle-hole bubble in

2D

Particle-hole bubble

Non-analyticities in the static limit (free

electrons)

Non-analyticities at small momentum and

frequency transfer

These non-analyticities in q correspond to

long-range correlations in real space (1/r2) and

can affect susceptibilities in a perturbation

expansion in the interaction U

Perturbative calculation of spin susceptibility

in a 2DEG

Consider screened Coulomb U and 2nd order pert.

theory in U

Chubukov, Maslov, PRB 68 ('03)

Nuclear magnetization at finite temperature.1

Magnon spectrum ?q becomes now linear in q due to

e-e interactions

(GaAs c20cm/s )

with spin wave velocity

Nuclear magnetization at finite temperature.2

where Tc is the 'Curie temperature'

? finite magnetization at finite temperature in

2D!

estimate for GaAs 2DEG Tc 25 µK

? temperatures are finite but still very small!

The local field factor approximation.1

with long history see e.g. Giuliani Vignale,

'06

Consider unscreened 2D-Coulomb interaction

Idea (Hubbard) replace the average electrostatic

potential seen by an electron by a local

potential

The local field factor approximation.2

?

i.e. again strong enhancement through

correlations

for

Giuliani Vignale, '06

strong enhancement of the Curie temperature

for rs 5-10

Conclusion

We use a Kondo lattice description (may suggest

numerical approach to attack nuclear spin

dynamics ?)

Electron-electron interactions permits a finite

Curie temperature

Electron-electron interactions increases the

Curie temperature

for large

Many open questions Disorder, nuclear spin

glass ? Spin decoherence in ordered phase?

Experimental signature?

Electron-electron interactions do matter to

determine the magnetic properties of 2D systems

i) Ferromagnetic semi-conductors ? ii) Some heavy

fermions materials ? iii) .

(No Transcript)

Experimental values for decay times in GaAs

quantum dots

Local Field Factor Approach

Idea replace the average electrostatic potential

by an effective local one

In the linear response regime, one may write

Hubbard proposal

Solve

Linear response

Towards a 2D nuclear spin model

y

x

where

at the mean field level

can reduce the quasi-2D problem to strictly 2D

lattice

Beyond simple perturbation theory.1

PS D Loss, PRL 2007 (cond-mat/0611292 )

vertex

see e.g. Giuliani Vignale, '06

G is the exact electron-hole scattering

amplitude and G the exact propagator

G obeys Bethe-Salpether equation as function of

p-h--irreducible vertex Girr

? solve Bethe-Salpether in lowest order in Girr

Beyond simple perturbation theory.2

PS D Loss, PRL 2007, (cond-mat/0611292)

Lowest approx. for vertex

? can derive simple formula

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