07/11/11 SCCS 2008

Sergey Kravchenko

in collaboration with:

PROFOUND EFFECTS OF ELECTRON-ELECTRON CORRELATIONS 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

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Why 2D?

Quantum Hall effect (Nobel Prize 1985)

High-Tc superconductors (Nobel Prize 1988)

FQHE (Nobel Prize 1998)

Graphene (Nobel Prize 2010)

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Outline

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

Samples

What do experiments show?

Magnetic properties of strongly coupled electrons in 2D: ballistic regime (no disorder)

Interplay between disorder and interactions in 2D; flow diagram

Conclusions

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-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

L

G = 1/R

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~1 ~35 rs

Gas Strongly correlated liquid Wigner crystal

Insulator ??????? Insulator

strength of interactions increases

Coulomb energyFermi energyrs =

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Suggested phase diagrams for strongly interacting electrons in two dimensions

Local moments, strong insulator

diso

rder

electron density

Local moments, strong insulator

diso

rder

electron density

Wignercrystal

Wignercrystal

Paramagnetic Fermi liquid, weak insulator Paramagnetic Fermi

liquid, weak insulator

Ferromagnetic Fermi liquid

Tanatar and Ceperley,

Phys. Rev. B 39, 5005 (1989)

Attaccalite et al.

Phys. Rev. Lett. 88, 256601 (2002)

strength of interactions increases strength of interactions increases

strongly disordered sample

clean sample

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

Samples

What do experiments show?

Magnetic properties of strongly coupled electrons in 2D: ballistic regime (no disorder)

Interplay between disorder and interactions in 2D; flow diagram

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)

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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

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

Samples

What do experiments show?

Magnetic properties of strongly coupled electrons in 2D: ballistic regime (no disorder)

Interplay between disorder and interactions in 2D; flow diagram

Conclusions

07/11/11

Strongly disordered Si MOSFET

(Pudalov et al.)

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

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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

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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)

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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:

07/11/11 SCCS 2008

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

(spins aligned)

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

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

07/11/11

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

Samples

What do experiments show?

Magnetic properties of strongly coupled electrons in 2D: ballistic regime (no disorder)

Interplay between disorder and interactions in 2D; flow diagram

Conclusions

07/11/11 Pisa 2006

How to study magnetic properties of 2D electrons?

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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

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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

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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

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-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

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0 1 2 3 4 5 6 7

d/d

B

1 B

B = 7 tesla

6 tesla

5 tesla

4 tesla

3 tesla

2 tesla

1.5 tesla

ns (1011 cm-2)

d/dB vs. ns in different parallel magnetic fields:

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Magnetic field of full spin polarization vs. electron density:

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5

magnetization data

magnetocapacitance data

linear fit

0

2

4

6

8

10 B

Bc (

me

V)

Bc (

tesl

a)

n

nc

electron density (1011 cm-2)

data become T-dependent

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Summary of the results obtained by four independent methods (including transport)

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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

insulator

T-dependent regime

Are we approaching a phase transition?

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diso

rder

electron density

Anderson insulator

paramagnetic Fermi-liquidWigner crystal?

Liquid ferromagnet?

Disorder increases at low density and we enter “Punnoose-

Finkelstein regime”

Density-independent disorder

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g-factor or effective mass?

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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

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Effective mass as a function of rs-2 in Si(111) and Si(100)

Si (111)

Si (100)

Shashkin, Kapustin, Deviatov, Dolgopolov, and Kvon, PRB (2007)

Si(111): peak mobility 2.5x103 cm2/Vs

Si(100): peak mobility 3x104 cm2/Vs

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

Samples

What do experiments show?

Magnetic properties of strongly coupled electrons in 2D: ballistic regime (no disorder)

Interplay between disorder and interactions in 2D; flow diagram

Conclusions

07/11/11 SCCS 2008

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

always insulating behavior

However, later this prediction was shown to be incorrect

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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

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Punnoose and Finkelstein, Science310, 289 (2005)

interactions

diso

rder

metallic phase stabilized by e-e interaction

disorder takes over

QCP

Recent development: two-loop RG theory

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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 testFirst, 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):

a

a

F

F

0

0

1 In standard Fermi-liquid notations,

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

S. Anissimova et al., Nature Phys. 3, 707 (2007)

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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

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

S. Anissimova et al., Nature Phys. 3, 707 (2007)

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

S. Anissimova et al., Nature Phys. 3, 707 (2007)

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Quantitative predictions of the one-loop RG for 2-valley systems(Punnoose and Finkelstein, Phys. Rev. Lett. 2002)

Solutions of the RG-equations for r << h/e2: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

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Resistance and interactions vs. T

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

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

S. Anissimova et al., Nature Phys. 3, 707 (2007)

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g-factor grows as T decreases

2.5

3

3.5

4

4.5

0 1 2 3 4

g*

T (K)

ns = 9.9 x 1010 cm-2

“ballistic” value

)1(2* g

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SUMMARY:

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

Disorder-interactions flow diagram of the metal-insulator transition clearly reveals a quantum critical point: i.e., there exists a metallic state and a metal-insulator transition in 2D, contrary to the 20-years old paradigm!