Nuclear Spin Ferromagnetic transition in a 2DEG

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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
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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

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I. THE HYPERFINE INTERACTION
I. SPIN FILTERING
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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)

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Hyperfine interaction for a single spin
Electron Zeeman energy
Hyperfine interaction
Nuclear spin dipole-dipole interaction
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Separation of the Hyperfine Hamiltonian
Hamiltonian
Note nuclear field
is a quantum operator
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Nuclear spins provide hyperfine field h with
quantum fluctuations seen by electron spin
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Nuclear spins provide hyperfine field h with
quantum fluctuations seen by electron spin
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Nuclear spins provide hyperfine field h with
quantum fluctuations seen by electron spin
With mean lthgt0 and quantum variance dh
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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
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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!
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II. NUCLEAR SPIN FERROMAGNETIC PHASE TRANSITION
IN A NON-INTERACTING 2D ELECTRON GAS ?
I. SPIN FILTERING
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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)
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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)
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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
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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
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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)).
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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
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The Weiss mean field theory.2
PS D Loss, PRL 2007
But is the simple MFT result really justified
for 2D ?
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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
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Susceptibility of the non-interacting 2DEG
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The 2D non-interacting electron gas
In the continuum limit
Electronic density in 2D
Expected and in agreement with the conjecture !
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III.Incorporating electron-electron interactions
I. SPIN FILTERING
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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
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Correction to spin susceptibility in 2nd order
in U
Chubukov Maslov, PRB 68, 155113 (2003)
(remaining diagrams cancel or give vanishing
contributions)
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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
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Perturbative calculation of spin susceptibility
in a 2DEG
Consider screened Coulomb U and 2nd order pert.
theory in U
Chubukov, Maslov, PRB 68 ('03)
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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
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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!
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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
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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
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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) .
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Experimental values for decay times in GaAs
quantum dots
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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
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Towards a 2D nuclear spin model
y
x
where
at the mean field level
can reduce the quasi-2D problem to strictly 2D
lattice
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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
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Beyond simple perturbation theory.2
PS D Loss, PRL 2007, (cond-mat/0611292)
Lowest approx. for vertex
? can derive simple formula