Persistent currents in two dimension: New regimes induced by the - - PowerPoint PPT Presentation

Persistent currents in two dimension: New regimes induced by the interplay between electronic correlations and disorder

Zoltán 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: few

interacting 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).

( ) [ ]

NRy r O r a r a E

s s s Fermi

× + + = ln

2 2 1

a1 = 2.0 a2 = –1.6972

• M. Gell-Mann and K. A. Brueckner
• Phys. Rev. 106, 364 (1957)

→

s

r

high density limit,

kinetic energy exchange energy

Strong interaction limit

• Electrons localized in real space (Wigner

crystal)

( ) [

]

NRy r O r c r c E

s s s Wigner

× + + =

− 2 2 / 3 2 / 3 1

• W. J. Carr Jr., Phys. Rev. 122, 1437

(1961) c1 = –2.21 c3/2 = 1.63

low density limit,

∞ →

s

r

classical elctrostatic energy quantum zero-point motion

• Fixed node Monte-Carlo

method:

• B. Tanatar and D. M. Ceperley

PRB 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 QMC

Density-density correlation function

Hybrid phase: nodal structure of Slater determinants in the crystal potential: mixed liquid-solid behavior

• H. Falakshahi, X. Waintal: Phys. Rev. Lett. 94, 046801 (2004)

rs

Theory for intermediate rs

Still 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 Hubbard model (high lattice filling, contact interaction).

Lattice model

N spinless fermions on L × L square lattice with periodic boundary conditions (lattice spacing s): s e U

2

=

2 2

2ms t h =

N r a a r

l B s

π 2 → = t UL r

l =

the t and U parameters of this Hamiltonian:

THE rs AND rl PARAMETERS: discrete Laplacian

j j j j i ij j i i,j j i

n W d n n U c c N- t ˆ ˆ ˆ 2 ˆ ˆ 4 H

∑

+ ∑ ⋅ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∑ ⋅ =

≠ > < +

ε

disorder

Persistent current

2 ) , 1 (

Im 2 ) ( Ψ Ψ = Φ

Φ + + L i j j long j

e c c j

π

) ( ) ( ) ( ~ 1 Φ − = Φ = Φ ∆ = ∑ E E E j L I

j long j long

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 real materials 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

y x eff

K L N K t E cos cos 2 2 π

( )

( )

y x eff

K K t E cos cos 2 + − = = Φ ( )

2 2

9 2 ~ L t E E I

eff lattice

π ≈ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ Φ = Φ − = Φ

where teff eff describes the rigid hopping of the three particle « molecule »

2 2 6 3

49 1296 π U L t teff =

Example: the persistent current

• t
• t
• t

∑ ⋅ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∑ ⋅ =

≠ > < + j i ij j i i,j j i

d n n U c c N- t 2 4 H

perturbation

rigid hopping

Ballsitic motion (BWM)

N=3 L=9 W=0.01 t=1 U=300

∑

+

− − =

' , '

4 H

j j j j eff Coul eff

C C t E Nt

3 3 1

~

− − N N N eff

L U t t

Effective Hamiltonian:

' j j C

C+ Density map Persistent current map

Coulomb Guided Stripes (CGS)

N=3 L=9 W=1 t=1 U=1000

1 3 / 2 3 / 2

) 2 cos( ~ ) ( ~

−

Φ Φ ∆

L L eff long

W t E I π

3 3 1

~

− − N N N eff

L U t t Disorder correction in the effective Hamiltonian:

∑ ∑

+ − − =

+ j j j j j j j eff Coul eff

N W C C t E Nt ˆ ' 4 H

' , '

ε

long trans

I I =

transverse current:

current of a collective motion

Localized Wigner Molceule (LWM)

N=3 L=6 W=20 t=1 U=1000

1 3 3

) 2 cos( ~ ) ( ~

− −

Φ Φ ∆

L L L long

U L t E I π Standard perturbation therory: current of independent particles

=

trans

I

transverse current:

Numerical check of the different regimes

ballistic localized stripe

L=6, t=1 L=6, W=1

localized stripe longitudinal longitudinal & transverse

Phase diagram for weak disorder

6 2 2

L U t t Wstripe ∝

2 / 3 2 / 1 2 / 1

L U t t Wloc ∝

localized stripe ballistic solid liq. 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

( )

⇒ + + ∇ + ∇ + ∇ − ≈ X M X r r h ˆ 2 H

2 3 2 2 2 1 2 el

E m

( ) ( )

2 6 2 5 2 4 2 3 6 1 2 2 2

4 10 2 H χ χ χ χ χ

α α

+ + + + + ∂ ∂ − ≈

∑

=

B B E m

el

h

3 2

24 6 D e B π =

( )

T L el

E E ω ω + + =

=

h

K

m B

L

20 = ω m B

T

8 = ω

Longitudinal modes Transverse modes 2nd order expansion around equilibrium where

D

Limit for the zero point motion

E0: ground state energy FN: scaling function

Lattice behavior:

Harmonic vibration of the solid in the continuum:

) , , ( ) , , ( ) , , ( ) , , ( t U L E t U L E t U L E t U L FN = = − =

N = 3 L = 6 L = 9

electrostatic energy l

r F 2327 .

3 =

Nt E E

el

4 = −

L = 12 L = 15 L = 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

• f 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 supersolid Sign 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 supersolid Sign 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

U Wglass ~ Crossover: The presence of a level crossing is specific to N.

Weak disorder

Phase diagram (N=3)

continuum lattice

localized stripe ballistic glass solid liq. hyb.

Conclusion

• In the presence of a weak disorder, we have

identified for large U/t three lattice regimes, characterized by different power-law decays as a function of U, t and W, L, N.

• The physics of the continuum is also

affected by disorder (signatures of the supersolid).