2/21/2007 NNCI 2007

Sveta Anissimova

Sergey Kravchenko(presenting author)

A. Punnoose

A. M. Finkelstein

Teun Klapwijk

Interplay between disorder and interactions in two dimensions

2/21/2007 NNCI 2007

-4

-3

-2

-1

0

1

3D

2D 1D

MIT

ln(G)

d(lnG)/d(lnL) = (G)

One-parameter scaling theory for non-interacting electrons: the origin of the common wisdom “all states are localized in 2D”

Abrahams, Anderson, Licciardello, and Ramakrishnan, PRL 42, 673 (1979)

G ~ Ld-2 exp(-L/Lloc)

metal (dG/dL>0)insulator

insulator

insulator (dG/dL<0)

Ohm’s law in d dimensions

QM interference

2/21/2007 NNCI 2007

However, the existence of the quantum Hall effect is inconsistent with this prediction

Solution (Pruisken, Khmelnitskii…):two-parameter (xx, xy) scaling theory

2/21/2007 NNCI 2007

Do the electron-electron interactions modify the“all states are localized in 2D at B=0” paradigm?

(what happens to the Anderson transition in the presence of interactions?)

2/21/2007 NNCI 2007

Corrections to conductivity due to electron-electron interactions in the diffusive regime (T < 1)

always insulating behavior

However, later this result was shown to be incorrect

2/21/2007 NNCI 2007

Zeitschrift fur Physik B (Condensed Matter) -- 1984 -- vol.56, no.3, pp. 189-96

Weak localization and Coulomb interaction in disordered systems

Finkel'stein, A.M. L.D. Landau Inst. for Theoretical Phys., Acad. of Sci., Moscow, USSR

0

02

2 1ln131ln

2 F

FT

e

Insulating behavior when interactions are weak Metallic behavior when interactions are strongEffective strength of interactions grows as the temperature decreases

Altshuler-Aronov-Lee’s result Finkelstein’s & Castellani-

DiCastro-Lee-Ma’s term

)45.01( 0 F)045.0( 0 F

2/21/2007 NNCI 2007

Same mechanism persists to ballistic regime (T> 1), but corrections become linear in temperature

where C(ns) < 0

This is reminiscent of earlier Stern-Das Sarma’s result

(However, Das Sarma’s calculations are not applicable to strongly interacting regime because at r s>1, the screening length becomes smaller than the separation between electrons.)

2/21/2007 NNCI 2007

What do experiments show?

2/21/2007 NNCI 2007

Strongly disordered Si MOSFET

(Pudalov et al.)

Consistent with the one-parameter scaling theory

2/21/2007 NNCI 2007

Kravchenko, Mason, Bowker, Furneaux, Pudalov, and D’Iorio, PRB 1995

Clean Si MOSFET, much lower electron densities

2/21/2007 NNCI 2007

Klapwijk’s sample: Pudalov’s sample:

In very clean samples, the transition is practically universal:

103

104

105

106

0 0.5 1 1.5 2

0.86x1011 cm-2

0.880.900.930.950.991.10

resi

stiv

ity

r (O

hm)

temperature T (K)

(Note: samples from different sources, measured in different labs)

2/21/2007 NNCI 2007

103

104

105

106

0 0.5 1 1.5 2

0.86x1011 cm-2

0.880.900.930.950.991.10

resistiv

ity r

(Ohm

)

temperature T (K)

clean sample: disordered sample:

… in contrast to strongly disordered samples:

Clearly, one-parameter scaling theory does not work here

2/21/2007 NNCI 2007

Again, two-parameter scaling theory comes to the rescue

2/21/2007 NNCI 2007

finite rincreases

while reduces r the interplay of disorder and r and interaction changes the trend and gives non-monotonic R(T)

2 22

2

2

22 2

11 ( ) ln(1 ) 1

(1

)

)

1

2

(4v v

dnn

d

d

d

r r

r

cooperon singlet “triplet”

ln(1/ ) , 1T T

Two parameter scaling

to all orders in 2

(Finkelstein, 1983-1984;Castellani, Di Castro, Lee, and Ma, 1984;Punnoose and Finkelstein, 2002; 2005)

2/21/2007 NNCI 2007

Punnoose and Finkelstein, Science310, 289 (2005)

interactions

diso

rde

r

metallic phase stabilized by e-e interaction

disorder takes over

QCP

2/21/2007 NNCI 2007

Low-field magnetoconductance in the diffusive regime yields strength of electron-electron interactions

1

2

Tk

Bg

B

B

22

22

2

1091.0

4,

T

B

k

g

h

eTB

B

B

Experimental test of the Punnoose-Finkelstein theoryFirst, one needs to ensure that the system is in the diffusive regime (T< 1).

One can distinguish between diffusive and ballistic regimes by studying magnetoconductance:

2

,

T

BTB

TBTB

2

,

- diffusive: low temperatures, higher disorder (Tt < 1).

- ballistic: low disorder, higher temperatures (Tt > 1).

The exact formula for magnetoconductance (Lee and Ramakrishnan, 1982):

2/21/2007 NNCI 2007

Experimental results (low-disordered Si MOSFETs; “just metallic” regime; ns= 9.14x1010 cm-2):

2/21/2007 NNCI 2007

Temperature dependences of the resistance (a) and strength of interactions (b)

This is the first time effective strength of interactions has been seen to depend on T

2/21/2007 NNCI 2007

Experimental disorder-interaction flow diagram of the 2D electron liquid

2/21/2007 NNCI 2007

Experimental vs. theoretical flow diagram(qualitative comparison b/c the 2-loop theory was developed for multi-valley systems)

2/21/2007 NNCI 2007

Quantitative predictions of the two-parameter scaling theory for 2-valley systems

(Punnoose and Finkelstein, Phys. Rev. Lett. 2002)

Solutions of the RG-equations:a series of non-monotonic curves r(T). After rescaling, the solutions are described by a single universal curve:

max

max max

ρ(T) = ρ R(η)

η = ρ ln(T /T)

r(T

)

(T)

rmax ln(T/Tmax)

Tmax

rmax

2 = 0.45

For a 2-valley system (like Si MOSFET),

metallic r(T) sets in when 2 > 0.45

2/21/2007 NNCI 2007

Resistance and interactions vs. T

Note that the metallic behavior sets in when 2 ~ 0.45, exactly as predicted by the RG theory

2/21/2007 NNCI 2007

Comparison between theory (lines) and experiment (symbols)(no adjustable parameters used!)

2/21/2007 NNCI 2007

Si-MOSFET vs. GaAs/AlGaAs heterostructures

Si-MOSFET advantages

Moderately high mobility: There exists a diffusive window T < 1/ < EF; 1/ = 2-3 K

Short range scattering: Anderson transition in a disordered Fermi Liquid (universal)

Two-valley system: Effects of electron-electron interactions are enhanced (“critical” 2=0.45 vs. 2.04 in a single-valley system)

GaAs/AlGaAs:

Ultra high mobility: Diffusive regime is hard to reach; 1/ < 100-200 mK

Long range scattering: Percolation type of the transition?

Very low density: Non-degeneracy effects; possible Wigner crystallization,..

2/21/2007 NNCI 2007

It is demonstrated, for the first time, that as a result of the interplay between the electron-electron interactions and disorder, not only the resistance but also the interaction strength exhibits a fan-like spread as the metal-insulator transition is crossed.

Conclusions:

Resistance-interaction flow diagram of the MIT clearly reveals a quantum critical point, as predicted by renormalization-group theory of Punnoose and Finkelstein.

The metallic side of this diagram is accurately described by the renormalization-group theory without any fitting parameters. In particular, the metallic temperature dependence of the resistance sets in once 2 > 0.45, which is in remarkable agreement with RG theory.

The interactions between electrons stabilize the metallic state in 2D and lead to the existence of a critical fixed point