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: T 2 =2T 1 (Golovach et al., PRL 93, (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 energy Nuclear Zeeman energy Hyperfine interaction Nuclear spin dipole-dipole interaction

Separation of the Hyperfine Hamiltonian Hamiltonian: Note: nuclear field longitudinal component flip-flop terms Separation: V V is a quantum operator

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

With mean =0 and quantum variance δh: Nuclear spins provide hyperfine field h with quantum fluctuations seen by electron spin:

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. 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 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/ )

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 Lacroix and Cyrot., PRB 20, 1969 (1979) A ferromagnetic phase expected at 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) Pure spin-spin Hamiltonian for nuclear spins only: 'RKKY interaction'

is the electronic longitudinal spin susceptibility in the static limit (ω=0). 'RKKY interaction ' where and Free electrons: J r 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 T c =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 assumeOne obtains a self-consistent mean field equation

For a 2D semiconductor with low electronic density n e << n must use Eq. (1): GaAs: The Curie temperature is still low! 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 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 !

I. SPIN FILTERING: III.Incorporating electron-electron interactions

Perturbative calculation of the spin susceptibility in a 2DEG Consider screened Coulomb U and 2 nd order pert. theory in U: Chubukov, Maslov, PRB 68, (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) Correction to spin susceptibility in 2 nd order in U: Chubukov & Maslov, PRB 68, (2003 ) 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 in real space (~1/r 2 ) and can affect susceptibilities in a perturbation expansion in the interaction U Non-analyticities at small momentum and frequency transfer:

Perturbative calculation of spin susceptibility in a 2DEG Consider screened Coulomb U and 2 nd order pert. theory in U: Chubukov, Maslov, PRB 68 ('03 ) where Γ s ~ - Um / 4π denotes the backscattering amplitude i.e. in the low q limit This linear -dependence (non-analyticity) permits ferromagnetic order with finite Curie temperature!

Nuclear magnetization at finite temperature.1 Magnon spectrum ω q becomes now linear in q due to e-e interactions: with spin wave velocity (GaAs: c~20cm/s ) What about q > 2k F ?  such q's are not relevant in m(T) for temperatures T with since then βω q >1 for all q>2k F

Nuclear magnetization at finite temperature.2 where T c is the 'Curie temperature': Note that self-consistency requires since aπ/a B ~1/10 in GaAs estimate for GaAs 2DEG: T c ~ 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 Determine 'local spin field factor' G - (q) semi-phenomenologically*: Thomas-Fermi wave vector, and g 0 =g(r=0) pair correlation function Note: G - (q) ~ q for q<2k F  this is in agreement also with Quantum Monte Carlo (Ceperley et al., '92,'95)

The local field factor approximation.2 Giuliani & Vignale, '06 strong enhancement of the Curie temperature:  i.e. again strong enhancement through correlations: for r s ~ 5-10 for

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 magnetic properties 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 Approach Idea: replace the average electrostatic potential by an effective local one In the linear response regime, one may write: Hubbard proposal: Linear response : 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 amplitude and G the exact propagator Γ obeys Bethe-Salpether equation as function of p-h--irreducible vertex Γ irr PS& D Loss, PRL 2007 (cond-mat/ ) vertex see e.g. Giuliani & Vignale, '06  solve Bethe-Salpether in lowest order in Γ irr

Beyond simple perturbation theory.2 Lowest approx. for vertex:  can derive simple formula: This leads to a dramatic enhancement of and therefore also of Curie temperature T c ~ δχ s onset of Stoner instability for Estimate: 'Stoner factor' as before use Maslov-Chubukov PS& D Loss, PRL 2007, (cond-mat/ )