Sergey Kravchenko

Approaching an (unknown) phase transition in two dimensions

A. Mokashi (Northeastern)

S. Li (City College of New York)

A. A. Shashkin (ISSP Chernogolovka)

V. T. Dolgopolov (ISSP Chernogolovka)

T. M. Klapwijk (TU Delft)

M. P. Sarachik (City College of New York)

in collaboration with:

~35 ~1 rs

Wigner crystal Strongly correlated liquid Gas

strength of interactions increases

Coulomb energy Fermi energyrs =

Terra incognita

Distorted lattice Short range order Random electrons

Suggested phase diagrams for strongly interacting electrons in two dimensions

strong insulator

diso

rder

electron density

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

clean sample

strongly disordered sample

University of Virginia

E C

EF

EF,

EC

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:

Why Si MOSFETs?

It turns out to be a very convenient 2D system to study strongly-interacting regime because of:

• Relatively large effective mass (0.19 m0 )

• Two valleys in the electronic spectrum

• Low average dielectric constant =7.7

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

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

For comparison, in n-GaAs/AlGaAs heterostructures, this would require 100 times lower electron densities. Such samples are not yet available.

10/10/09 University of Virginia

Al

SiO2 p-Si

2D electrons conductance band

valence band

chemical potential

+ _

ener

gy

distance into the sample (perpendicular to the surface)

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

Metal-insulator transition in 2D semiconductors

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 (

Ohm

)

temperature T (K)

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

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

104

105

106

107

108

109

1010

0 1 2 3 4 5

r (W

)

H|| (Tesla)

Shashkin et al., 2000

Si MOSFET

T = 35 mK

MITn

s just above the zero-field MIT

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 1.2T (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!

Spin susceptibility near nc

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)

Magnetoresistance in a parallel magnetic field

Shashkin, Kravchenko, Dolgopolov, and

Klapwijk, PRL 2001

Bc

Bc

Bc

0

1

2

3

4

5

0 2 4 6 8 10

BB c (meV

)

ns (1015 m-2)

nc

0

0.2

0.4

0.8 1.2 1.6 2

BB c (meV

)

ns (1015 m-2)

Extrapolated polarization field, Bc, vanishes at a finite electron density, n

Shashkin, Kravchenko, Dolgopolov, and

Klapwijk, PRL 2001

Spontaneous spin polarization at n?

n

gm as a function of electron densitycalculated using g*m* = ћ2ns / BcB

1

2

3

4

5

0 2 4 6 8 10

gm/g

0mb

ns (1015 m-2)

ns= n

c

(Shashkin et al., PRL 2001)

n

2D electron gas Ohmic contact

SiO2

Si

Gate

Modulated magnetic fieldB + B

Current amplifierVg

+

-

Magnetic measurements without magnetometer

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

-15 A

)

ns (1011 cm-2)

1 fA!!

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

B|| = 5 tesla

Bar-Ilan University

Raw magnetization data: induced current vs. gate voltageIntegral of the previous slide gives M (ns):complete spin polarization

ns (1011 cm-2)

M (1

011 B

/cm

2 )

met

al

insu

lato

r

0

0.5

1

1.5

0 2 4 6

B|| = 5 tesla

at ns=1.5x1011 cm-2

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 datamagnetocapacitance dataintegral of the master curvetransport data

/ 0

ns (1011 cm-2)

nc

insulator

T-dependent regime

g-factor or effective mass?

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

0

1

2

3

4

0 2 4 6 8 10

m/m

b , g

/g0

ns (1011 cm-2)

g/g0

m/mb

Effective mass vs. g-factor (from the analysis of the transport data in spirit of

Zala, Narozhny, and Aleiner, PRB 2001) :

Another way to measure m*: amplitude of the weak-field Shubnikov-de Haas oscillations

vs. temperature

(Rahimi, Anissimova, Sakr, Kravchenko, and Klapwijk, PRL 2003)

250

300

350

400

0.2 0.25 0.3 0.35 0.4 0.45 0.5

r (W

/squ

are)

B_|_ (tesla)

430 mK

230 mK

42 mK

1000

2000

3000

4000

0 0.2 0.4 0.6 0.8 1r

(W/s

quar

e)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

Comparison of the effective masses determined by two independent experimental methods:

0

1

2

3

4

0 1 2 3 4

m/m

b

ns (1011 cm-2)

50 30 20 15 12r

s

(Shashkin, Rahimi, Anissimova, Kravchenko, Dolgopolov, and

Klapwijk, PRL 2003)

15 11 8 rs

Thermopower

Thermopower : S = - V / (T) S = Sd + Sg = T + Ts

V : heat either end of the sample, measure the induced voltage difference in the shaded region

T : use two thermometers to determine the temperature gradient

Divergence of thermopower

1/S tends to vanish at nt

Critical behavior of thermopower

(-T/S) ∝ (ns – nt)x where x = 1.0±0.1

nt=(7.8±0.1)×1010 cm-2

and is independent of the level of the disorder

In the low-temperature metallic regime, the diffusion thermopower of strongly interacting 2D electrons is given by the relation

S/T ∝ m/ns

Therefore, divergence of the thermopower indicates a divergenceof the effective mass:

m ∝ ns /(ns − nt)

(Dolgopolov and Gold, 2011)

We observe the increase of the effective mass up to m 25mb 5me!!

i. using Gutzwiller's theory (Dolgopolov, JETP Lett. 2002)

ii. using an analogy with He3 near the onset of Wigner crystallization (Spivak and Kivelson, PRB 2004)

iii. solving an extended Hubbard model using dynamical mean-field theory (Pankov and Dobrosavljevic, PRB 2008)

iv. from a renormalization group analysis for multi-valley 2D systems (Punnoose and Finkelstein, Science 2005)

v. by Monte-Carlo simulations (Marchi et al., PRB 2009; Fleury and Waintal, PRB 2010)

A divergence of the effective mass has been predicted…

What is the nature of the low-density phase?

Transport properties

If the insulating state were due to a single-particle localization, the electric field needed to destroy it would be of order (the most conservative estimate)

Eth ~ Wb /le ~ 103 – 104 V/m

However, in experimentEth = 1 – 10 V/m !

De-pinning of a pinned Wigner solid?

Differential resistivity, dV/dI

Broadband noise

Transport properties of the insulating phase favor pinned Wigner solidformation

SUMMARY:

• In the clean regime, spin susceptibility critically grows upon approaching to some sample-independent critical point, n, pointing to the existence of a phase transition.

• The dramatic increase of the spin susceptibility is due to the divergence of the effective mass rather than that of the g-factor and, therefore, is not related to the Stoner instability. It may be a precursor phase or a direct transition to the long sought-after Wigner solid.

• However, the existing data, although consistent with the formation of the Wigner solid, are not enough to reliably confirm its existence.