* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Nuclear Spin Ferromagnetic transition in a 2DEG Pascal Simon
Aharonov–Bohm effect wikipedia , lookup
Quantum state wikipedia , lookup
Canonical quantization wikipedia , lookup
Hydrogen atom wikipedia , lookup
Scalar field theory wikipedia , lookup
Electron configuration wikipedia , lookup
EPR paradox wikipedia , lookup
Electron scattering wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Renormalization wikipedia , lookup
Franck–Condon principle wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Nuclear force wikipedia , lookup
Nitrogen-vacancy center wikipedia , lookup
Molecular Hamiltonian wikipedia , lookup
Bell's theorem wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Electron paramagnetic resonance wikipedia , lookup
History of quantum field theory wikipedia , lookup
Spin (physics) wikipedia , lookup
Ising model wikipedia , lookup
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 I. SPIN FILTERING: INTERACTION 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 (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) Hyperfine interaction for a single spin Electron Zeeman energy b g B * B z Nuclear Zeeman energy gI N 3 * 10 g B Hyperfine interaction Nuclear spin dipole-dipole interaction Separation of the Hyperfine Hamiltonian Hamiltonian: H g B BS z S h H 0 V Note: nuclear field h Ai I i is a quantum operator i Separation: H 0 ( g B B hz ) S z V 1 h S h S 2 h hx ih y longitudinal component flip-flop terms ... ... V ... V ... Nuclear spins provide hyperfine field h with quantum fluctuations seen by electron spin: S h Nuclear spins provide hyperfine field h with quantum fluctuations seen by electron spin: S h Nuclear spins provide hyperfine field h with quantum fluctuations seen by electron spin: h S With mean <h>=0 and quantum variance δh: h h2 nucl 2 Ak I k k 1 N A / N 5mT (10ns) 1 nucl 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. S Ii S Ii E 0 Total suppression requires full polarization of nuclear spins which is not currently achievable Polarization of nuclear spins 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? Problem is quite old and was first studied in 1940 by Fröhlich & Nabarro for bulk metals! II. NUCLEAR SPIN FERROMAGNETIC I.PHASE SPIN FILTERING: 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/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: H eff A2 8n I q q (q)I q 'RKKY interaction' n N /V (justified since nuclear spin dynamics is much slower than electron dynamics) Assuming no electronic polarization: (q) s (q) An effective nuclear spin Hamiltonian H eff A2 8n 1 s (q) I q I q J r r' I r I r' 2 r,r' q J r r' where and A2 s (r r ' ) 8n 'RKKY interaction' s (r ) zz (r , 0) is the electronic longitudinal spin susceptibility in the static limit (ω=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 I ( I 1) A2 Tc Ne 12 n PS & D Loss, PRL 2007 For a 2D semiconductor with low electronic density ne << n must use Eq. (1): GaAs: A 90eV , N e ne / EF Tc 5K The Curie temperature is still low! 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 I. SPIN FILTERING: 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 Correction to spin susceptibility in 2nd order in U: 0 (q' , ) correction to self-energy Σ(q,ω) Chubukov & Maslov, PRB 68, 155113 (2003) 2 2 d kd q' dd 2 s (q) 8U (2 ) 6 G02 (k , )G0 (k q, )G0 (k q' , ) 0 (q' , ) (remaining diagrams cancel or give vanishing contributions) Non-analyticities in the particle-hole bubble in 2D Particle-hole bubble: d 2 pdm (q, n ) G( p, m ) G( p q, m n ) 3 (2 ) Non-analyticities in the static limit (free electrons): ( p q , m n ) ( p, m ) m* 0 (q,0) , for q 2k F 2 m* q 2k F 0 (q ~ 2k F ,0) (1 ) , for q 2k F 2 kF Non-analyticities at small momentum and frequency transfer: n m* 0 ( q, n ) (1 ) 2 2 2 (v F q ) n 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: 0 (q' , ) Chubukov, Maslov, PRB 68 ('03) s (q) 4 q s (0)s2 / 3k F , q 2k F , i.e. in the low q limit where Γs ~ - Um / 4π denotes the backscattering amplitude q -dependence (non-analyticity) permits This linear ferromagnetic order with finite Curie temperature! Nuclear magnetization at finite temperature.1 1 q m(T ) I dq q 2n 0 e 1 Magnon spectrum ωq becomes now linear in q due to e-e interactions: IA2 q s (q) c q , 2n with spin wave velocity for q 2k F I A2 N 0 c ( N 0U ) 2 nk F 4 12 (GaAs: c~20cm/s ) What about q > 2kF ? such q's are not relevant in m(T) for temperatures T with T T2 k F c 2k F / k B since then βωq>1 for all q>2kF Nuclear magnetization at finite temperature.2 T2 1 q m(T ) I dq cq I 1 2 , 2n 0 e 1 Tc where Tc is the 'Curie temperature': T T2 k F c Tc 2 kB 3nI finite magnetization at finite temperature in 2D! estimate for GaAs 2DEG: Tc ~ 25 μK Note that self-consistency requires T T2 k F Tc temperatures are finite but still very small! 2 a Tc 3I aB rs since aπ/aB~1/10 in GaAs The local field factor approximation.1 with long history: see e.g. Giuliani & Vignale*, '06 Consider unscreened 2D-Coulomb interaction V (q) 2e 2 / q Idea (Hubbard): replace the average electrostatic potential seen by an electron by a local potential: s (q) 0 (q) 1 V (q )G (q ) 0 (q ) Determine 'local spin field factor' G-(q) semi-phenomenologically*: G (q) g 0 q q g 0 2 (1 p / s ) 1 2 2 / aB Thomas-Fermi wave vector, and g0=g(r=0) pair correlation function 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 s (q) 0 (q) 1 V (q )G (q ) 0 (q ) , G (q) g 0 s (q) N 0 q (1 s / p ) 2 2 q q g 0 2 (1 p / s ) 1 g0 i.e. again strong enhancement through correlations: (1 s / p ) 2 ~ 10 1 / g 0 (rs ) e1.46rs ~ 103 for rs aB 1 / n ~ 5 Giuliani & Vignale, '06 strong enhancement of the Curie temperature: Tc ~ O(mK ) 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) …. 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: Solve Linear response : s (q) 0 (q) 1 V (q )G (q ) 0 (q ) , 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 ) s (q) vertex i L2 ' G ( p q)G ( p) p ' p, , ' p ' ' i L2 p , p ' ' G ' ( p' )G ' ( p' q) p' see e.g. Giuliani & Vignale, '06 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 Γirr solve Bethe-Salpether in lowest order in Γirr Beyond simple perturbation theory.2 PS& D Loss, PRL 2007, (cond-mat/0611292) Lowest approx. for vertex: irr (q, ) U can derive simple formula: s (q) 1 (q) q (1 U(q)) 2 q as before use Maslov-Chubukov (q) 0 (0) N e onset of Stoner instability for This leads to a dramatic enhancement of s (q) and therefore also of Curie temperature Tc ~ Estimate: δχs Tc ~ 25 (1 Um / ) 2 K O(mK ) 'Stoner factor' UNe ~ 1 (rs 1)