Sergey Kravchenko

transcript

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:

• Low disorder: weak (logarithmic) insulator

Non-interacting electron gas in two dimensions (boring):

• High disorder: strong (exponential) insulator

~35 ~1 rs

Wigner crystal Strongly correlated liquid Gas

strength of interactions increases

Coulomb energy Fermi energyrs =

Terra incognita

Suggested phase diagrams for strongly interacting electrons in two dimensions

strong insulator

electron density

strong insulator

electron density

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

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:

• large effective mass m*= 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

SiO2 p-Si

2D electrons conductance band

valence band

chemical potential

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:

0 0.5 1 1.5 2

0.86x1011 cm-2

0.880.900.930.950.991.10

temperature T (K)

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

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

(Hanein, Shahar, Tsui et al., PRL 1998)

Similar transition has later been observed in other 2D structures:

• p-Si:Ge (Coleridge’s group; Ensslin’s group)

• p-GaAs/AlGaAs (Tsui’s group, Boebinger’s group)

• n-GaAs/AlGaAs (Tsui’s group, Stormer’s group, Eisenstein’s group)

• n-Si:Ge (Okamoto’s group, Tsui’s group)

• p-AlAs (Shayegan’s group)

0 1 2 3 4 5

H|| (Tesla)

Shashkin et al., 2000

Si MOSFET

T = 35 mK

s just above the zero-field MIT

The effect of the parallel magnetic field:

0 0.3 0.6 0.9 1.2

0.7650.7800.7950.8100.825

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

0 2 4 6 8 10 12

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

0 2 4 6 8 10

BB c (meV

ns (1015 m-2)

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?

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

0 2 4 6 8 10

ns (1015 m-2)

(Shashkin et al., PRL 2001)

2D electron gas Ohmic contact

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

Magnetization of non-interacting electrons

spin-down spin-up

Magnetic field of the full spin polarization vs. ns

Bc = h2ns/2Bmb

M = Bns =Bns B/Bc for B < Bc

Bns for B > Bc

B > Bc

B < Bc

Bc = h2ns/B g*m*

non-interacting systemspontaneous spin polarization at n

0 1 2 3 4 5 6 7

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)

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)

0.5 1 1.5 2 2.5 3 3.5

magnetization datamagnetocapacitance dataintegral of the master curvetransport data

ns (1011 cm-2)

insulator

T-dependent regime

g-factor or effective mass?

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

0 2 4 6 8 10

ns (1011 cm-2)

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)

0.2 0.25 0.3 0.35 0.4 0.45 0.5

B_|_ (tesla)

430 mK

230 mK

0 0.2 0.4 0.6 0.8 1r

_|_ (tesla)

T = 42 mK

0.3 0.4 0.5 0.6

132 mK

42 mK82 mK

high density low density

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

0 1 2 3 4

ns (1011 cm-2)

50 30 20 15 12r

(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

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

T/S ∝ ns /m

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

m ∝ ns /(ns − nt)

Divergence of the effective electron mass

Dolgopolov and Gold, JETP Lett. 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, PRB 2003; Spivak and Kivelson, PRB 2004)

iii. extending the Fermi liquid concept to the strongly-interacting limit (Khodel et al., PRB 2008)

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

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

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

A divergence of the effective mass has been predicted…

Transport properties of the insulating phase

If the insulating state were due to a single-partical localization, a severalorders of magnitude higher electric field would be needed to delocalize an electron:

eEc l > Wb ~ 0.1 – 1 meV

But if this is a pinned Wigner solid,the electric field “pulls” many electrons while only one is pinned – hence, much weaker electric fields are required to depin it and break localization.

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. Unfortunately, residual disorder does not allow to see this transition in currently available samples

• 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.

Sergey Kravchenko

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