Transcript Nuclear Spin Ferromagnetic transition in a 2DEG Pascal Simon

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  90eV , Ne  ne / EF
Tc  5K
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' dd
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 pdm
 ( 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 / 3k 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

2n 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  ,

2n 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)  2e 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)