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Approaching an (unknown) phase transition in two dimensions. Sergey Kravchenko. in collaboration with:. 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)

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slide1

Approaching an (unknown) phase transition

in two dimensions

Sergey Kravchenko

in collaboration with:

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)

slide2

Coulomb energy

Fermi energy

rs =

Wigner crystal Strongly correlated liquid Gas

strength of interactions increases

~35 ~1 rs

Terra incognita

Distorted lattice

Short range order

Random electrons

slide3

Suggested phase diagrams for strongly interacting electrons in two dimensions

strong insulator

strong insulator

disorder

disorder

electron density

electron density

Attaccalite et al. Phys. Rev. Lett. 88, 256601 (2002)

Tanatar and Ceperley, Phys. Rev. B 39, 5005 (1989)

strongly disordered sample

Wigner

crystal

Ferromagnetic Fermi liquid

Paramagnetic Fermi liquid, weak insulator

Wigner

crystal

Paramagnetic Fermi liquid, weak insulator

clean sample

strength of interactions increases

strength of interactions increases

slide4

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:

University of Virginia

slide5

WhySi 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)
  • Twovalleys 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.

slide6

Al

SiO2

p-Si

conductance band

2D electrons

chemical potential

energy

valence band

_

+

distance into the sample (perpendicular to the surface)

10/10/09

University of Virginia

slide7

Metal-insulator transition in 2D semiconductors

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

slide8

In very clean samples, the transition is practically universal:

Sarachik and Kravchenko, PNAS 1999;

Kravchenko and Klapwijk, PRL 2000

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

slide10

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

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

(spins aligned)

slide12

Magnetoresistancein a parallel magnetic field

T = 30 mK

Bc

Bc

Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRL 2001

Bc

Spins become fully polarized

(Okamoto et al., PRL 1999;

Vitkalov et al., PRL 2000)

slide13

Extrapolated polarization field, Bc,

vanishes at a finite electron density, n

Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRL 2001

n

Spontaneous spin polarization at n?

gm as a function of electron density calculated using g m 2 n s b c b
gm as a function of electron densitycalculated using g*m* = ћ2ns / BcB

(Shashkin et al., PRL 2001)

n

slide15

Magnetic measurements without magnetometer

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

1010 Ohm

-

+

Gate

Current amplifier

Vg

SiO2

Modulated magnetic field

B + B

Si

2D electron gas

Ohmic contact

i∝d/dB = - dM/dns

slide17

Raw magnetization data: induced current vs. gate voltage

Integral of the previous slide gives M (ns):

complete spin polarization

at ns=1.5x1011 cm-2

B|| = 5 tesla

Bar-Ilan University

slide18

Spin susceptibility exhibits critical behavior near the

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

insulator

T-dependent

regime

shashkin kravchenko dolgopolov and klapwijk prb 2002

Effective mass vs. g-factor

(from the analysis of the transport data in spirit of

Zala, Narozhny, and Aleiner, PRB 2001) :

Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRB (2002)
slide21

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

high density

low density

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

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

15 11 8

rs

(Shashkin, Rahimi, Anissimova, Kravchenko, Dolgopolov, and Klapwijk, PRL 2003)

slide24

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

slide27

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

slide28

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 divergence

of the effective mass:

m ∝ ns /(ns − nt)

(Dolgopolov and Gold, 2011)

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

slide29

A divergence of the effective mass has been predicted…

  • using Gutzwiller's theory (Dolgopolov, JETP Lett. 2002)
  • using an analogy with He3 near the onset of Wigner crystallization (Spivakand Kivelson, PRB 2004)
  • solving an extended Hubbard model using dynamical mean-field theory (Pankov and Dobrosavljevic, PRB 2008)
  • from a renormalization group analysis for multi-valley 2D systems (Punnoose and Finkelstein, Science 2005)
  • by Monte-Carlo simulations (Marchiet al., PRB 2009; Fleury and Waintal, PRB 2010)
slide31

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 experiment

Eth =1 – 10 V/m !

De-pinning of a pinned Wigner

solid?

slide32

Differential resistivity, dV/dI

Broadband noise

Transport properties of the insulating phase favor pinned Wigner solid

formation

slide33

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.