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

I. THE HYPERFINE INTERACTION

II. NUCLEAR SPIN FERROMAGNETIC PHASE TRANSITION

IN A NON-INTERACTING 2D ELECTRON GAS ?

III. INCORPORATING ELECTRON-ELECTRON INTERACTIONS

IV. CONCLUSION

OUTLOOK

I. SPIN FILTERING: I. THE HYPERFINE INTERACTION

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 (Khaetskii&Nazarov, PRB’00) and: T2=2T1 (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; Coish&Loss, PRB2004)

Central issue for quantum computing: decoherence of spin qubit

Hyperfine interaction for a single spin

Electron Zeeman energyNuclear Zeeman energy

3

*10

B

NI

g

g

Hyperfineinteraction

Nuclear spin dipole-dipole interaction

zBBgb *

Separation of the Hyperfine Hamiltonian

Hamiltonian: VHhSBSgH zB 0

i

ii IAh

Note: nuclear field

... ... ... ...

zzB ShBgH )(0

ShShV2

1

longitudinal component

flip-flop terms

Separation:

yx ihhh V V

is a quantum operator

Nuclear spins provide hyperfine field h with quantum fluctuations seen by electron spin:

S

h

S

h

Nuclear spins provide hyperfine field h with quantum fluctuations seen by electron spin:

With mean <h>=0 and quantum variance δh:

1

2

1

2 )10(5/

nsmTNAIAhh

nucl

N

kkknucl

Sh

Nuclear spins provide hyperfine field h with quantum fluctuations seen by electron spin:

Suppression due to a high magnetic field

S

iI

•The hyperfine interaction is suppressed in the presence of a magnetic field (electron Zeeman splitting) since electron spin – nuclear spin flip-flops do notconserve energy.

S

iI

0E

Total suppression requires full polarization of nuclear spins which is not currently

achievable

1. Dynamical polarization

• optical pumping: <65%, 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?

Polarization of nuclear spins

Problem is quite old and was first studied in 1940 by Fröhlich & Nabarro for bulk metals!

I. SPIN FILTERING:

II. NUCLEAR SPIN FERROMAGNETIC

PHASE TRANSITION IN A NON-INTERACTING

2D ELECTRON GAS ?

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 previousdescription 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 exactlyand corresponds to a ferromagnetic spin state

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

Several elaborated mean field theoryhave been used to obtain the phase Diagram of the 3D Kondo lattice

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

A ferromagnetic phase expectedat small A/t and low electronic density ?

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)

Assuming no electronic polarization:

(justified since nuclear spin dynamics is much slower than electron dynamics)

qq

qeff IqIn

AH

)(8

2

Pure spin-spin Hamiltonian for nuclear spins only:

VNn /

'RKKY interaction'

)()( qq s

is the electronic longitudinal spin susceptibility in the static limit (ω=0).

'r'r,r

r'rrqq

qseff IIJIIqn

AH

2

1)(

8

2

'RKKY interaction'

)0,()( rr zzs

)(8

2

'rrn

AJ s'rr

where

and

Free electrons: Jr is standard RKKY interaction Ruderman & Kittel, 1954

Note that result is also valid in the presence of electron-electron interactions

An effective nuclear spin Hamiltonian

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

ec Nn

AIIT

2

12

)1(

For a 2D semiconductor with low electronic density ne << n must use Eq. (1):

GaAs:

KTc 5 The Curie temperature is still low!

Fee EnNeVA /,90

PS & D Loss, PRL 2007

But: is the simple MFT result really justified for 2D ?

The Weiss mean field theory.2

Spin wave calculations

The mean field calculations and other results on the 3D Kondo lattice suggesta 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 !

I. SPIN FILTERING: III.Incorporating electron-electron

interactions

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 Π (sharp Fermi surface)

non-Fermi liquid behavior in 2D

(remaining diagrams cancel or give vanishing contributions)

),'(),(),(

)2(

'8)(

0020

6

222

qkqkk GGG

ddqkddUqs

),'(0 q

Correction to spin susceptibility in 2nd order in U:

Chubukov & Maslov, PRB 68, 155113 (2003)

),'(0 q

correction to self-energy Σ(q,ω)

Non-analyticities in the particle-hole bubble in 2D

Non-analyticities in the static limit (free electrons):

Particle-hole bubble:

These non-analyticities in q correspond to long-range correlations inreal space (~1/r2) and can affect susceptibilities in a perturbation expansion in the interaction U

Non-analyticities at small momentum and frequency transfer:

),(),()2(

),(3

2

nmmm

n qpGpGpdd

q

Fkqform

q 2,2

)0,(*

0

FF

FF kqfor

k

kqmkq 2,)

21(

2)0,2~(

*

0

))(

1(2

),(22

*

0

nF

nn

qv

mq

),( nmqp

),( mp

Perturbative calculation of spin susceptibility in a 2DEG

Consider screened Coulomb U and 2nd order pert. theory in U:

Chubukov, Maslov, PRB 68 ('03)

where Γs ~ - Um / 4π denotes the backscattering amplitude

i.e. in the low q limit,2,3/)0(4)( 2FFsss kqkqq

),'(0 q

This linear -dependence (non-analyticity) permits ferromagnetic order with finite Curie temperature!

q

Nuclear magnetization at finite temperature.1

12

1)(

0

qe

qdq

nITm

Fsq kqforqcqn

IA2,)(

2

2

Magnon spectrum ωq becomes now linear in q due to e-e interactions:

20

02

)(124

UNNA

nk

Ic

F with spin wave velocity (GaAs: c~20cm/s )

BFk kkcTTF

/22

What about q > 2kF ? such q's are not relevant in m(T) for temperatures T with

since then βωq>1 for all q>2kF

Nuclear magnetization at finite temperature.2

,112

1)(

2

2

0

ccq T

TI

e

qdq

nITm

nI

k

cT

Bc

32where Tc is the 'Curie temperature':

csB

ck Tra

a

ITTT

F

3

22

Note that self-consistency requires

since aπ/aB~1/10 in GaAs

FkTT 2

estimate for GaAs 2DEG: Tc ~ 25 μK

finite magnetization at finite temperature in 2D!

temperatures are finite but still very small!

The local field factor approximation.1

Consider unscreened 2D-Coulomb interaction

Idea (Hubbard): replace the average electrostatic potential seen by an electron by a local potential:

with long history: see e.g. Giuliani & Vignale*, '06

qeqV /2)( 2

)()()(1

)()(

0

0

qqGqV

qqs

Determine 'local spin field factor' G-(q) semi-phenomenologically*:

Thomas-Fermi wave vector,and g0=g(r=0) pair correlationfunction

120

0 )/1()(

spgq

qgqG

Ba/22

Note: G-(q) ~ q for q<2kF this is in agreement also with Quantum Monte Carlo (Ceperley et al., '92,'95)

The local field factor approximation.2

Giuliani & Vignale, '06

,)()()(1

)()(

0

0

qqGqV

qqs

120

0 )/1()(

spgq

qgqG

strong enhancement of the Curie temperature:

0

2

20

)/1()(

g

qNq ps

s

i.e. again strong enhancement through correlations:

346.10

2

10~)(/1

10~)/1(

srs

ps

erg

)(~ mKOTc for rs ~ 5-10

for 5~/1 nar Bs

Conclusion

Electron-electron interactions permits a finite Curie temperature

Electron-electron interactions increases the Curie temperature

for large

Electron-electron interactions do matter to determine the magneticproperties of 2D systems

i) Ferromagnetic semi-conductors ?ii) Some heavy fermions materials ?iii) ….

We use a Kondo lattice description (may suggest numerical approach to attack nuclear spin dynamics ?)

Many open questions: Disorder, nuclear spin glass ? Spin decoherence in ordered phase? Experimental signature?

Experimental values for decay times in GaAs quantum dots

charge

Local Field Factor ApproachIdea: replace the average electrostatic potential by an effective local one

In the linear response regime, one may write:

Hubbard proposal:

Linear response :

,)()()(1

)()(

0

0

qqGqV

qqs

Solve

Towards a 2D nuclear spin model

x

y

where

at the mean field level:

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

Beyond simple perturbation theory.1

Γ is the exact electron-hole scattering amplitudeand G the exact propagator

Γ obeys Bethe-Salpether equation as function of p-h--irreducible vertex Γirr

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

vertex

see e.g. Giuliani & Vignale, '06

'',,

2)()(')(

p

ps pGqpG

L

iq

)'()'( ''

''',2'' qpGpGL

i

pppp

solve Bethe-Salpether in lowest order in Γirr

Beyond simple perturbation theory.2

Lowest approx. for vertex: can derive simple formula:Uqirr ),(

q

q

qUq

qs

)(

))(1(

1)(2

eNq )0()( 0

This leads to a dramatic enhancement of

and therefore also of Curie temperature Tc ~ δχs

)(qs

onset of Stoner instability for 1~eUN

)1( sr

Estimate: )()/1(25~ 2 mKOKUmTc

'Stoner factor'

as before useMaslov-Chubukov

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