persistent currents in two dimension: new regimes induced by the …coqusy06/slides/nemeth.pdf ·...
TRANSCRIPT
Persistent currents in two dimension: New regimes induced by the interplay between electronic
correlations and disorderZoltán Ádám Németh
Jean-Louis Pichard
CEA - Saclay, Service de Physique de l'Etat Condensé
Outline:
• Overview of the strongly correlated 2D electron-gas problem;
• Introduction of the numerical model: fewinteracting particles on a lattice
• Persistent current maps with disorder
References:
Z.Á. Németh and J.-L. Pichard, Eur. Phys. J. B 45, 111 (2005)
J.-L. Pichard and Z.Á. Németh, J. Phys. IV France 131, 155 (2005)
• Fermi: weakly-interacting quantum particles• Wigner: strongly interacting particles
Quantum solid state physics
TWO LENGTH SCALES:• average interparticle spacing a• Bohr-radius aB
Dimensionless scaling parameter: rs= a/aB
Weak interaction limit
• Electrons localized in k-space (Fermi liquid behavior).
( ) [ ]NRyrOra
raE s
ss
Fermi ×++= ln221
0
a1 = 2.0a2 = –1.6972
M. Gell-Mann and K. A. BruecknerPhys. Rev. 106, 364 (1957)
0→srhigh density limit,kinetic energy
exchange energy
Strong interaction limit
• Electrons localized in real space (Wigner crystal)
( ) [ ]NRyrOrc
rcE s
ss
Wigner ×++= − 22/32/31
0
W. J. Carr Jr., Phys. Rev. 122, 1437 (1961)
c1 = –2.21c3/2 = 1.63
low density limit, ∞→srclassical elctrostatic energy
quantum zero-point motion
• Fixed node Monte-Carlo method:
B. Tanatar and D. M. CeperleyPRB 39, 5005 (1989)
Quantum Monte-Carlo
solid
polarized liquid
rs
GS energy
rs = 37±5
unpolarized liquid
Semiconductor heterostructures
Since the ’70s it is possible to fabricate 2D electron gas in semiconductor devices.
Electron density and rs are varied through voltage gates.Example:
quantum point-contact
Unexpected metallic behavior in 2D
In ultra-clean heterostructures:rs can reach ≈ 40
Observed metallic behavior at intermediate rs
• S.V. Kravchenko et al., PRB 50, 8039 (1994)
• J. Yoon et al., PRL 82, 1744 (1999)
resistivity
temperature
low density
intermediate density
Hybrid phase in QMCDensity-density correlation function
Hybrid phase: nodal structure of
Slater determinants in the crystal potential:
mixed liquid-solid behaviorH. Falakshahi, X. Waintal: Phys. Rev. Lett. 94, 046801 (2004)
rs
Theory for intermediate rsStill mainly speculations...
• Andreev-Lifshitz « supersolid » state (relation with He-physics)
• Inhomogeneous phases, stripes and bubbles (B. Spivak)
Relation with the physics of high-Tc cuprates and with Hubbardmodel (high lattice filling, contact interaction).
Lattice modelN spinless fermions on L × L square lattice with periodic
boundary conditions (lattice spacing s):
seU
2
=2
2
2mst h
=N
raar l
Bs π2
→=t
ULrl =
the t and U parameters of this Hamiltonian:
THE rs AND rl PARAMETERS:
discrete Laplacian
jj
jji
ij
ji
i,jji nW
dnnUccN-t ˆˆˆ
2ˆˆ4H ∑+∑⋅+⎟
⎠⎞
⎜⎝⎛ ∑⋅=
≠ ><
+ ε
disorder
Persistent current
02
)0,1(0Im2)( ΨΨ=Φ Φ+
+ Lijj
longj eccj π
)()0()(~1000 Φ−=Φ=Φ∆= ∑ EEEj
LI
j
longjlong
Longitudinal and transverse currents
•local:
•total:
Transverse current is analogous.
Persistent currents with disorder
• Effects of an infinitesimal disorder: new lattice perturbative regime
– Ballistic motion– Coulomb Guided Stripes– Localization if the Wigner crystal
Can be relevant in realmaterials
e.g. Cobalt-oxides(NaxCoO2)
• effective mass: m*/m = 175• relative dielectric constant: εr = 20• lattice spacing s = 2.85 Å• carrier density depends on Na+
concentration
Lemmens et al., cond-mat/0309186
Strong interaction, lattice regimes
Lattice perturbation theory when t → 0
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎠⎞
⎜⎝⎛ +−=⎟
⎠⎞⎜
⎝⎛ Φ=Φ yxeff K
LNKtE coscos22
0 π
( ) ( )yxeff KKtE coscos20 +−==Φ
( ) 2
20 9
20~L
tEEI efflatticeπ
≈⎟⎠⎞⎜
⎝⎛ Φ=Φ−=Φ
where teffeff describes the rigid hopping of the three particle« molecule »
22
63
491296
πULtteff =
Example: the persistent current
-t
-t
-t∑⋅+⎟⎠⎞
⎜⎝⎛ ∑⋅=
≠ ><
+
jiij
ji
i,jji d
nnUccN-t2
4H
perturbation
rigid hopping
Ballsitic motion (BWM)
N=3 L=9 W=0.01 t=1 U=300
∑ +−−=',
'4Hjj
jjeffCouleff CCtENt
331~ −
−N
N
N
eff LU
tt
Effective Hamiltonian:
'jj CC+
Density mapPersistent current map
Coulomb Guided Stripes (CGS)
N=3 L=9 W=1 t=1 U=1000
13/2
3/2
0
)2cos(~)(~ −
ΦΦ∆ L
Leff
long Wt
EIπ
331~ −
−N
N
N
eff LU
tt
Disorder correction in the effective Hamiltonian:
∑∑ +−−= +
jjj
jjjjeffCouleff NWCCtENt ˆ'4H
',' ε
longtrans II =transverse current:
current of a collective motion
Localized Wigner Molceule (LWM)
N=3 L=6 W=20 t=1 U=1000
1
33
0)2cos(~)(~ −
− ΦΦ∆ L
LL
long ULtEI π
Standard perturbation therory:
current of independent particles
0=transItransverse current:
Numerical check of the different regimes
ballistic
localized
stripe
L=6, t=1 L=6, W=1
localized
stripe
longitudinallongitudinal & transverse
Phase diagram for weak disorder
62
2
LUt
tWstripe ∝
2/32/1
2/1
LUt
tWloc ∝
localized
stripe
ballistic
solidliq. hyb.
Critical lines:
continuum lattice
In case of long-range interaction, lattice models without disorder
exhibit a latice-continuum transition.
Continuum perturbation theory when rs → ∞
Zero point motion in the vibrating mode of the molecule
( ) ⇒++∇+∇+∇−≈ XMXrrh ˆ
2H 2
322
21
2
elEm
( ) ( )26
25
24
23
6
12
22
4102
H χχχχχα α
+++++∂∂
−≈ ∑=
BBEm elh
3
2
246
DeB π
=
( )TLelEE ωω ++== h0K mB
L20
=ωmB
T8
=ω
Longitudinal modes Transverse modes
2nd order expansion around equilibrium
where
D
Limit for the zero point motion
E0: ground state energyFN: scaling function
Lattice behavior:
Harmonic vibration of the solid in the continuum:
),0,()0,,(),,(),,(
0
00
tULEtULEtULEtULFN =
=−=
N = 3L = 6
L = 9
electrostatic energy
lrF 2327.03 =
NtEE el 40 =−
L = 12L = 15L = 18
rl = UL/t
Example:Three spinless fermions on L × L lattice
Persistent currents with disorder
Do we see similar thing with disorder?
• Effect of an intermediate disorder in the continuum limit:– Coulomb guided stripes– Level crossing and supersolid behavior
N=3 L=9 W=1 t=1 U=50
Parallel density and current
Continuum version
of the Coulomb
guided stripe
N=3 L=9 W=1 t=1 U=1000
Coulomb guided
stripe on a lattice
N=3 L=9 W=1 t=1 U=7
Legett’s rule:
1D motion diamagnetic
means even number of
particles
Sign of supersolidSign of supersolid
N=3 L=9 W=1 t=1 U=15
Crossover regime
Disconnected current
and density
N=3 L=9 W=1 t=1 U=7
Legett’s rule:
1D motion diamagnetic
means even number of
particles
Sign of supersolidSign of supersolid
N=3 L=9 W=1 t=1 U=15
Crossover regime
Disconnected current
and density
Level crossing & strong disorder
paramagnetic
diamagentic
Strong disorder
UWglass ~Crossover:
The presence of a
level crossing is
specific to N.
Weak disorder
Phase diagram (N=3)
continuum lattice
localized
stripe
ballistic
glass
solidliq. hyb.
Conclusion
• In the presence of a weak disorder, we haveidentified for large U/t three lattice regimes,characterized by different power-law decaysas a function of U, t and W, L, N.
• The physics of the continuum is also affected by disorder (signatures of the supersolid).