Sveta AnissimovaAnanth Venkatesan (now at UBC)Mohammed Sakr (now at UCLA)Mariam Rahimi (now at UC Berkeley)Sergey Kravchenko

Alexander ShashkinValeri Dolgopolov

Teun Klapwijk

Silicon MOSFETs GaAs/AlGaAs heterostructures SiGe heterostructures Surface of a material (liquid helium, graphene sheets)

10

1

sF

ee

s

nE

Er

EC

EF

EF,

EC

electron density

At low densities, ns ~ 1011 cm-2, Coulomb energy exceeds Fermi energy:

EC >> EF

meVne

E sC 102

meVnm

E sF 58.02 *

2

electron density decreases

strength of interactions increases

rs = EC / EF >10 – strongly interacting regime can easily be reached

large m* = 0.19 m0

average = 7.7 two valleys nv = 2

Hanein, Shahar, Tsui et al., PRL 1998Kravchenko, Mason, Bowker, Furneaux, Pudalov, and D’Iorio, PRB 1995

sVcm 24104Similar 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)

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)

103

104

105

106

0 0.5 1 1.5 2

0.86x1011 cm-2

0.880.900.930.950.991.10

resistiv

ity r

(Ohm

)

temperature T (K)

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

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

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

- Diagonal resistance

- Hall resistance

xxR

xyR

• 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

IRV

IRV

xyH

xxxx

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

(Okamoto et al., PRL 1999; Vitkalov et al., PRL 2000)

Shashkin, Kravchenko, Dolgopolov, Klapwijk, PRL 2001

Bc

Bc

Bc

0

1

2

3

4

5

6

0 2 4 6 8 10 12

BB

c (meV

)

ns (1015 m-2)

nc

Shashkin et al, 2001

Vitkalov, Sarachik et al, 2001

Pudalov et al, 2002

n

Vanishing Bc at a finite n nc indicates a ferromagnetic transition in this electron system

The fact that n is sample independent and n nc indicates that the MIT in clean samples is driven by interactions

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

gm as a function of electron density calculated using

1

2

3

4

5

0 2 4 6 8 10

gm/g

0m

b

ns (1015 m-2)

ns= n

c

Shashkin et al., PRL 2001

n

cB

s

B

nmg

2

**

Effective Mass Measurements: 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

/sq

uare

)

B_|_ (tesla)

430 mK

230 mK

42 mK

1000

2000

3000

4000

0 0.2 0.4 0.6 0.8 1r

(W/s

qua

re)

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:ns = 5x1011 cm-2 ns = 1.2x1011 cm-2

dots – ν = 10

squares – ν = 14

solid line – fit by L-K formula

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.

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

ns = 1.2x1011 cm-2

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 12

rs

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

Therefore, the sharp increaseof the spin susceptibility nearthe critical density is due to theenhancement of the effective mass rather then g-factor, unlike in the Stoner scenario

*

2D electron layer

Ohmic contact

SiO2

Si

Gate

Modulated magnetic fieldB + Bmod

Current-to-Voltage converterVg

+

-

Measurements of thermodynamic magnetization

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

;2 mod

dB

d

e

fCBiM

C – capacitance - chemical potential sdn

dM

dB

d

Maxwell relation:

R=1010 W

Lock-inamplifier

LVC6044 CMOS Quad Micropower OperationalAmplifier with noise level: 0.2 fA/(Hz)1/2

f = 0.45 HzBmod = 0.01 – 0.03 tesla

Magnetic field of the full spin polarization Bc vs. ns

Bc

ns0

Bc = h2ns/2Bmb

ns

0

dMdns

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

Bns for B > Bc

B > Bc

B < Bc

B

Bc = h2ns/B g*m*

n

non-interacting systemspontaneous spin polarization at n

-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

the onset of completespin polarization

d/dB = 0

Shashkin, Anissimova, Sakr, Kravchenko,Dolgopolov, and Klapwijk, cond-mat/0409100

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

complete spin polarization

ns (1011 cm-2)

M (

101

1 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

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

Shashkin, Anissimova, Sakr, Kravchenko,Dolgopolov, and Klapwijk, cond-mat/0409100

-0.2

-0.1

0

0.1

0.2

0 1 2 3 4 5 6 7

7 T6 T5 T

4 T3 T2 T

ns (1011 cm-2)

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.5 1 1.5 2 2.5

magnetization data

linear fit

0

2

4

6

8

10 B

Bc (

meV

)

ns (1011 cm-2)

Bc (

tesl

a)

n

nc

Spontaneous spin polarization at n?

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

2D electron layerOhmic contact

SiO2

Si

Gate

Modulated gate voltageVg + Vg

Current-to-Voltage converterVg

+

-

Measurements of thermodynamic density of states

sdn

d

AeCC

2

0

111

C0 – geometric capacitanceA – sample area

R=1010 W

Lock-in amplifier

LVC6044 CMOS Quad Micropower OperationalAmplifier with noise level: 0.2 fA/(Hz)1/2

f = 0.3 HzVg = 0.09VC0 = 624 pF

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

0

0.002

0.004

0.006

0.008

1 2 3 4

[C(0

) -

C(B

)] /

C(0

)

9.9 tesla

ns (1011 cm-2)

9 tesla

8 tesla

7 tesla

6 tesla

5 tesla

4 tesla

D-1

ns

fully spin-polarized electrons

spin-unpolarized electrons

Polarization field from capacitance measurements:

Shashkin, Anissimova, Sakr, Kravchenko,Dolgopolov, and Klapwijk, cond-mat/0409100

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, possibly due to localized band-tail

Shashkin, Anissimova, Sakr, Kravchenko,Dolgopolov, and Klapwijk, cond-mat/0409100

Spin susceptibility exhibits critical behavior near the

metal-insulator transition: ~ 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

cannot measure

Shashkin, Anissimova, Sakr, Kravchenko,Dolgopolov, and Klapwijk, cond-mat/0409100

d/dB vs. ns in perpendicular magnetic field

-300

-250

-200

-150

-100

-50

0

50

0 1 2 3 4 5 6 7 8

-120

-100

-80

-60

-40

-20

0

20

0 1 2 3 4 5 6

i (1

0-1

5 A)

d/

dB

( B

)

ns (1011 cm-2)

out-of-phase

in-phase

g-factor measurementsin perpendicular fields:

-20

-15

-10

-5

0

5

10

15

20

1 1.5 2 2.5 3 3.5

-8

-6

-4

-2

0

2

4

6

8

1 1.5 2 2.5 3 3.5 4i (

10-1

5 A)

gate voltage (volts)

d/

dB

(B)

ns (1011 cm-2)

"spin-up"

"spin-down"

BB

gB

*

Anissimova, Venkatesan, Shashkin, Sakr, Kravchenko, and Klapwijk, cond-mat/0503123

g-factor:

-20

-10

0

10

20

0 0.2 0.4 0.6

-5

0

5

10

g* = 2.23"spin-up"

"spin-down"

(a)

i (10

-15 A

)

d/d

B ( B

)

B = 6 tesla

-20

-10

0

10

20

-5

0

5

0 0.2 0.4 0.6

B = 5 tesla

g* = 2.34

"spin-down"

"spin-up"

(b)

i (10

-15 A

)

d/d

B ( B

)

-40

-20

0

20

40

0 0.1 0.2 0.3 0.4 0.5

-10

0

10

g* = 2.2

i (10

-15 A

) "spin-down"

"spin-up"

(c)

d/d

B ( B

)

|-2|

B = 3.5 tesla-8

-4

0

4

8

0 0.2 0.4 0.6

-15

-10

-5

0

5

10(d)

|-2|

N = 1

N = 02(m

e/m*) cos = 2.2

B = 8 tesla

= 66.4o

i (10

-15 A

)

d/d

B ( B

)

g-factor and effective mass:

Anissimova, Venkatesan, Shashkin, Sakr, Kravchenko, and Klapwijk, cond-mat/0503123

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

0

1

2

3

4

5

6

7

8

0.5 1.5 2.5 3.5

g*/

2, m

*/m b

, and

/

0

ns (1011 cm-2)

nc g*/2

/0

m*/mb

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)

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

Insulating behavior when interactions are weak Metallic behavior when interactions are strong Magnetic field destroys metal

0

02

2 1ln131ln

2 F

FT

e

Insulating behavior when interactions are weak

Metallic behavior when interactions are strong

Magnetic field destroys metal

0

1

2

3

4

5

6

7

8

0.5 1.5 2.5 3.5

g*/

2, m

*/m b

, and

/

0

ns (1011 cm-2)

nc g*/2

/0

m*/mb

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

These conclusions are in agreement with experiments

Punnoose and Finkelstein, Science, Vol. 310. no. 5746, pp. 289 - 291

Punnoose and Finkelstein, ScienceVol. 310. no. 5746, pp. 289 - 291

• 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