Metal-Insulator Transition in 2D Electron Systems
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Metal-Insulator Transition in 2D Electron Systems. Sveta Anissimova Ananth Venkatesan (now at UBC) Mohammed Sakr (now at UCLA) Mariam Rahimi (now at UC Berkeley) Sergey Kravchenko Alexander Shashkin Valeri Dolgopolov Teun Klapwijk.

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Sveta Anissimova Ananth Venkatesan (now at UBC) Mohammed Sakr (now at UCLA)

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Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Metal-Insulator Transition in 2D Electron Systems

Sveta Anissimova

Ananth Venkatesan (now at UBC)

Mohammed Sakr (now at UCLA)

Mariam Rahimi (now at UC Berkeley)

Sergey Kravchenko

Alexander Shashkin

Valeri Dolgopolov

Teun Klapwijk


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Si MOSFET: Silicon Metal-Oxide-Semiconductor Field-Effect Transistor

2-D QUANTUM SYSTEMS:

  • Silicon MOSFETs

  • GaAs/AlGaAs heterostructures

  • SiGe heterostructures

  • Surface of a material (liquid helium, graphene sheets)


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Why Si MOSFET?

  • largem* =0.19 m0

  • average = 7.7

  • twovalleysnv = 2

At low densities,

ns ~ 1011 cm-2,

Coulomb energy exceeds Fermi energy: EC >> EF

electron density decreases

strength of interactions increases

rs = EC / EF >10 – strongly interacting

regimecan easily be reached


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Metal-Insulator Transition in 2D

High-Mobility Si MOSFETs

  • Similar transition is also observed in other 2D structures:

  • p-Si:Ge (Coleridge’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)

B = 0

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

Hanein, Shahar, Tsui et al., PRL 1998


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

In very clean samples, the transition is practically universal:

Klapwijk’s sample:

Pudalov’s sample:

(Note: samples from

different sources,

measured in different labs)


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

… in contrast to strongly disordered samples:

disordered sample:

clean sample:


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

The effect of magnetic field


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

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

(spins aligned)


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

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


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

How to study magnetic properties of

2D electrons?


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Transport measurements: standard four-terminal technique

- Diagonal resistance

- Hall resistance

  • Rotator equipped Oxford dilution refrigerator

  • Base temperature ~ 30 mK

  • High mobility (100)-Si MOSFET μ=3 m2/Vs at T=0.1 K

  • Excitation current 0.1 – 0.2 nA

  • f = 0.4 Hz


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Magnetoresistance in a parallel magnetic field:

T = 30 mK

Bc

Bc

Bc

Spins become fully polarized

Shashkin, Kravchenko, Dolgopolov, Klapwijk, PRL 2001

(Okamoto et al., PRL 1999;

Vitkalov et al., PRL 2000)


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Extrapolated polarization field, Bc,

vanishes at a finite electron density, nc

Vanishing Bc at a

finite n  ncindicates aferromagnetic transition

in this electron system

The fact that n is sample independent and n  nc indicates that the MIT in clean samplesis drivenby interactions

Shashkin et al, 2001

Pudalov et al, 2002

Vitkalov, Sarachik et al, 2001

nc


Gm as a function of electron density calculated using

~ gm as a function of electron density calculated using

Shashkin et al., PRL 2001

nc


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Effective Mass Measurements: amplitude of the weak-field Shubnikov-de Haas oscillations vs. temperature

high density:

ns= 5x1011 cm-2

low density:

ns = 1.2x1011 cm-2

Rahimi, Anissimova, Sakr,

Kravchenko, and Klapwijk, PRL 2003


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

dots – ν = 10

squares – ν = 14

solid line – fit by L-K formula

ns = 1.2x1011 cm-2

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

The amplitude of the SdH oscillations follows the calculated curve down to the lowest achieved temperature: the electrons are in a good thermal contact with the bath.


Comparison of the effective masses determined by two independent experimental methods

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

*

Therefore, the sharp increase

of the spin susceptibility near

the critical density is due to the

enhancement of the effective massrather then g-factor, unlike in the Stoner scenario

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


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Measurements of thermodynamic magnetization

suggested by B. Halperin (1998); first implemented by Prus et al. (2003)

LVC6044 CMOS Quad Micropower Operational

Amplifier with noise level: 0.2 fA/(Hz)1/2

f = 0.45 Hz

Bmod = 0.01 – 0.03 tesla

R=1010 

-

Lock-in

amplifier

+

Vg

Gate

Current-to-Voltage converter

SiO2

Modulated magnetic field

B + Bmod

Si

2D electron layer

Ohmic contact

Maxwell relation:

C – capacitance

 - chemical potential


Magnetic field of the full spin polarization b c vs n s

Magnetic field of the full spin polarization Bc vs. ns

spontaneous spin polarization at nc:

non-interacting system

mBns B/Bc forB < Bc

Bc = ph2ns/mB g*m*

Bc = ph2ns/2mBmb

M = mBx ns =

mBns forB > Bc

dM

Bc

dns

B > Bc

0

B

ns

nc

B < Bc

0

ns


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Raw magnetization data: induced current vs. gate voltage

dm/dB = - dM/dn

B|| = 5 tesla

1 fA!!

the onset of complete

spin polarization

d/dB = 0

Shashkin, Anissimova, Sakr, Kravchenko,

Dolgopolov, and Klapwijk, cond-mat/0409100


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

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


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

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

Shashkin, Anissimova, Sakr, Kravchenko,

Dolgopolov, and Klapwijk, cond-mat/0409100


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Magnetic field of full spin polarization vs.electron density from magnetization measurements

Spontaneous spin polarization at nc?


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Measurements of thermodynamic density of states

LVC6044 CMOS Quad Micropower Operational

Amplifier with noise level: 0.2 fA/(Hz)1/2

f = 0.3 Hz

Vg = 0.09V

C0 = 624 pF

R=1010 

-

Lock-in

amplifier

+

Vg

Gate

Current-to-Voltage converter

SiO2

Si

2D electron layer

Ohmic contact

Modulated gate voltage

Vg + dVg

C0– geometric capacitance

A – sample area


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Polarization field from capacitance measurements:

Jump in the density of states signals the onset of full spin polarization

D-1

fully spin-polarized electrons

ns

spin-unpolarized electrons

Shashkin, Anissimova, Sakr, Kravchenko,

Dolgopolov, and Klapwijk, cond-mat/0409100


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Magnetic field of full spin polarization vs. electron density:

data become T-dependent, possibly due to localized band-tail

electron density (1011 cm-2)

Shashkin, Anissimova, Sakr, Kravchenko,

Dolgopolov, and Klapwijk, cond-mat/0409100


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Spin susceptibility exhibits critical behavior near the

metal-insulator transition: c ~ ns/(ns – nc)

insulator

cannot measure

Shashkin, Anissimova, Sakr, Kravchenko,

Dolgopolov, and Klapwijk, cond-mat/0409100


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

g-factor or effective mass?


D m d b vs n s in perpendicular magnetic field

dm/dB vs. ns in perpendicular magnetic field


G factor measurements in perpendicular fields

g-factor measurementsin perpendicular fields:

Anissimova, Venkatesan, Shashkin, Sakr,

Kravchenko, and Klapwijk, cond-mat/0503123


G factor

g-factor and effective mass:

g-factor:

Anissimova, Venkatesan, Shashkin, Sakr,

Kravchenko, and Klapwijk, cond-mat/0503123


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Summary of the results obtained by four (or five) independent methods


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

  • spin susceptibility critically grows near the metal-insulator transition

  • the enhancement of the g-factor is weak and practically density independent

  • the effective mass becomes strongly enhanced as the density is decreased

Shashkin, Anissimova, Sakr, Kravchenko, Dolgopolov, and Klapwijk, Phys. Rev. Lett. 96, 036403 (2006);

Anissimova, Venkatesan, Shashkin, Sakr, Kravchenko, and Klapwijk, Phys. Rev. Lett. 96, 046409 (2006)

Shashkin, Rahimi, Anissimova, Kravchenko, Dolgopolov, and Klapwijk, Phys. Rev. Lett. 91, 046403 (2003)


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Temperature-dependent corrections to conductivity due to

electron-electron interactions

Low temperatures, Diffusive regime

(Tt<<1)


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

  • 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

(always “insulating” behavior)

  • Insulating behavior when interactions are weak

  • Metallic behavior when interactions are strong

  • Magnetic field destroys metal


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Same corrections persist to ballistic regime

(Higher temperatures; Tt>>1)

  • Insulating behavior when interactions are weak

  • Metallic behavior when interactions are strong

  • Magnetic field destroys metal


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Theory of the metal-insulator transition

(diffusive regime)


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

…the point of the metal to insulator transition correlates with the appearance of the divergence in the spinsusceptibility… note that at the fixed point the g-factor remains finite

Punnoose and Finkelstein, Science,

Vol. 310. no. 5746, pp. 289 - 291

These conclusions are in agreement with experiments


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

Punnoose and Finkelstein, Science

Vol. 310. no. 5746, pp. 289 - 291


Sveta anissimova ananth venkatesan now at ubc mohammed sakr now at ucla

SUMMARY:

  • Pauli spin susceptibility critically grows with a tendency to diverge near the critical electron density

  • We find no sign of increasing g-factor, but the effective mass is strongly (×3) enhanced near the metal-insulator transition

and…

Punnoose-Finkelstein theory gives a quantitatively correct description of the metal-insulator transition in 2D


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