07/11/11 SCCS 2008

Sergey Kravchenko

in collaboration with:

AMAZING PROPERTIES OF STRONGLY CORRELATED ELECTRONS IN TWO DIMENSIONS

A. Punnoose M. P. Sarachik A. A. Shashkin CCNY CCNY ISSP

S. Anissimova V. T. Dolgopolov A. M. Finkelstein T. M. KlapwijkNEU ISSP Texas A&M TU Delft

Outline

Scaling theory of localization: “all electrons are localized in 2D”

Samples

What do experiments show?

Magnetic properties of strongly correlated electrons in 2D

Conclusions

Band theory of metals:

E F

Conduction band

Valence band

Conduction band

Insulator Metal

But it turns out that even if the Fermi level lies in the conduction band,the system may be insulating.

Localization by disorder (Anderson localization)

Low disorder High disorder

Electrons can penetrate infinitely far(with some scattering)

Electrons are localized at a certain distance, called “localization length”, Lloc

Lloc

“… very few believed in [localization] at the time, and even fewer saw its importance... It has yet to receive adequate mathematical treatment, and one has to resort to the indignity of numerical simulations to settle even the simplest questions about it."

P.W. Anderson, Nobel Lecture, 1977

In 1979, a powerful theory was created by the “Gang of Four” (Abrahams, Anderson, Licciardello, and Ramakrishnan), according to

which, there is no conductivity in 2D at low temperatures.

This became one of the most influential paradigms in modern condensed matter physics.

However, this prediction is valid for non-interacting electrons only.

~1 ~35 rs

Gas Strongly correlated liquid Wigner crystal

Insulator Insulator

strength of interactions increases

Coulomb energy Fermi energyrs =

Terra incognita

But electrons do interact via Coulomb forces!

University of Virginia

EC

EF

EF, E

C

electron density

In 2D, the kinetic (Fermi) energy is proportional to the electron density:

EF = (h2/m) Ns

while the potential (Coulomb) energy is proportional to Ns1/2:

EC = (e2/ε) Ns1/2

Therefore, the relative strength of interactions increases as the density decreases:

Scaling theory of localization: “all electrons are localized in two dimensions

Samples

What do experiments show?

Magnetic properties of strongly correlated electrons in 2D

Conclusions

07/11/11 SCCS 2008

silicon MOSFETAl

SiO2 p-Si

2D electrons conductance band

valence band

chemical potential

+ _

ener

gy

distance into the sample (perpendicular to the surface)

SCCS 2008

Why Si MOSFETs?

• large m*= 0.19 m0

• two valleys

• low average dielectric constant =7.7

As a result, at low electron densities, Coulomb energy strongly exceeds Fermi energy: EC >> EF

rs = EC / EF >10 can easily be reached in clean samples

EC

EF

EF, E

C

electron density

Scaling theory of localization: “all electrons are localized in two dimensions

Samples

What do experiments show?

Magnetic properties of strongly correlated electrons in 2D

Conclusions

Strongly disordered Si MOSFET

(Pudalov et al.)

Consistent (more or less) with the one-parameter scaling theory

S.V. Kravchenko, G.V. Kravchenko, W. Mason, J. Furneaux, V.M. Pudalov, and M. D’Iorio, PRB 1995

Clean sample, much lower electron densities

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)

tem perature T (K )

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

Sarachik and Kravchenko, PNAS 1999;Kravchenko and Klapwijk, PRL 2000

103

104

105

0 2 4 6 8 10 12

r (O

hm)

B (Tesla)

1.01x1015 m-2

1.20x1015

3.18x1015

2.40x1015

1.68x1015

T = 30 mK

Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRL 2001

The effect of the parallel magnetic field:

104

105

106

0 0.3 0.6 0.9 1.2

r (W

)

T (K)

B = 0

0.7650.7800.7950.8100.825

104

105

106

0 0.3 0.6 0.9 1.2

T (K)

1.0951.1251.1551.1851.215

B > Bsat

Shashkin et al., 2000

Magnetic field, by aligning spins, changes metallic R(T) to insulating:

Such a dramatic reaction on parallel magnetic field suggests unusual spin properties!

Scaling theory of localization: “all electrons are localized in 2D”

Samples

What do experiments show?

Magnetic properties of strongly correlated electrons in 2D

Conclusions

103

104

105

0 2 4 6 8 10 12

r (O

hm)

B (Tesla)

1.01x1015 m-2

1.20x1015

3.18x1015

2.40x1015

1.68x1015

T = 30 mK

Spins become fully polarized (Okamoto et al., PRL 1999; Vitkalov et al., PRL 2000)

Method 1: magnetoresistance in a parallel magnetic field

Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRL 2001

Bc

Bc

Bc

Method 2: weak-field Shubnikov-de Haas oscillations

(Pudalov et al., PRL 2002; Shashkin et al, PRL 2003)

250

300

350

400

0.2 0.25 0.3 0.35 0.4 0.45 0.5

r (W

/sq

uare

)

B_|_ (tesla)

430 mK

230 mK

42 mK

1000

2000

3000

4000

0 0.2 0.4 0.6 0.8 1

r (W

/sq

uare

)

B_|_ (tesla)

T = 42 mK

2800

2900

3000

3100

0.3 0.4 0.5 0.6

132 mK

42 mK82 mK

=14

=10

= 6

high density low density

2D electron gas Ohmic contact

SiO2

Si

Gate

Modulated magnetic fieldB + B

Current amplifierVg

+

-

Method 3: measurements of thermodynamic magnetization

suggested by B. I. Halperin (1998); first implemented by O. Prus, M. Reznikov, U. Sivan et al. (2002)

i ~ d/dB = - dM/dns

1010 Ohm

-2

-1

0

1

2

0 1 2 3 4 5 6 7

-1

-0.5

0

0.5

1

d/d

B ( B

)

i (10

-15A

)

ns (1011 cm-2)

1 fA!!

Raw magnetization data: induced current vs. gate voltaged/dB = - dM/dn

B|| = 5 tesla

Spin susceptibility exhibits critical behavior near the

sample-independent critical density n : ~ ns/(ns – n)

1

2

3

4

5

6

7

0.5 1 1.5 2 2.5 3 3.5

magnetization data

magnetocapacitance data

integral of the master curve

transport data

/ 0

ns (1011 cm-2)

nc

Are we approaching a phase transition?

g-factor or effective mass?

Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRB 66, 073303 (2002)

0

1

2

3

4

0 2 4 6 8 10

m/m

b ,

g/

g 0

ns (1011 cm-2)

g/g0

m/mb

Effective mass vs. g-factor

Not the Stoner scenario! Wigner crystal? Maybe, but evidence is insufficient

SUMMARY:

(i) There exists a metallic state in 2D, contrary to the 30-years old paradigm!

(ii) Strong interactions in clean two-dimensional systems lead to strong increase and possible divergence of the spin susceptibility: the

behavior characteristic of a phase transition

(iii) The dramatic increase of the spin susceptibility is caused by the effective mass rather than by the g-factor