Sasha Kuntsevich

Nimrod Teneh

Vladimir Pudalov

Spin-droplet state of an interacting 2D electron system

M. Reznikov

• Magnetic order in clean low-density systems

• Methods of magnetization measurements

Recharging Technique• Experimental results• Implications

Technion

Electron gas with interactions

Short range repulsive interaction2nd order phase transition into ferromagnetic ordered state

0

1 U

Interactions characterized by with

2

sF F

ee

E Eer

For a single-valley system

Stoner (1947)

Stoner instability

Ferromagnetic Bloch Instability

Hartree-Fock approximationUnscreened interaction, no

correlations: ~ 2

Decreasing density

1 2 3 n n n

Ener

gy

Long range ineraction

n nn

Phase diagramAttaccalite et al. (2001)

First order transition at rs~20: Senatore et al. (2001)

rs~26

Clean system

• Very low density Wigner Crystal rs~37

B. Tanatar and D.C. Ceperley (1989)

ferromagnetic

Clean system

Very small energy difference!

antiferromagnetic

ferromagnetic

• Very low density Wigner Crystal rs~37

B. Tanatar and D.C. Ceperley (1989)

Methods: Shubnikov - de Haas beatings

F. Fang and P. Stiles (1968), T. Okamoto at al., (1999), S. Vitkalov at al. (2000), V. Pudalov at.al., (2001)

2 4 6 7rs

V. Pudalov at al, (2001)

Metal-Insulator Transition in a Silicon Inversion Layer

m

gmBB

In-plane magnetoresistance

S. Vitkalov et al. PRL 2001A. Shashkin et al. PLR, 2001

In-plane magnetoresistance

A. Shashkin et al. PLR, 2001

Possible FM transition ??

Samples: Si Field effect transistors

Russian samples, beginning of 80th, Holland samples, mid 80th

1/ 2 11 2/ 8 (at 10 )s Bn r a n cm

Typical parameters

ps

Valley degeneracy 2 therefore

m3.4 x104 cm2/Vs @1.7K5 [mm]

The Principle of the Recharging Technique

, m B

f fn B

mn

mm

Maxwell relation:

0 2D Gcn Ve e

m m

2 0 2D GD c VcedndB e B B e

m m m

0 0 22

0 0

1 1 1 Dc cnc c n c e n

m - geometrical capacitance

magnetic moment per unit area

Important: / /, so the recharging method is distinct from magnetocapacitance.

Small correction

Finite thickness contributions to at

Diamagnetic contribution

Capacitance contribution

change

Recharging Technique

_+

VG

Out

Modulated magnetic field

B+dB

Current Amplifier

Ohmic contact

Gate

SiO2

Si

2D electron gas

( )  ( ) V Bi CI i C V I B

e B m d d

𝜕𝑚𝜕𝑛 =− 𝜕𝜇

𝜕𝐵𝐶 (𝜔)

can be measured whenever is measurable

i.e. recharging technique is applicable even in the insulator!

Expected behaviorT=0, finite magnetic field

m

gmBB

Interactions

M

n

No interactions

n

Mn

mB

No interactions

Interactions Prus et al,2003 B>T

B (T)

/

/ at n=1.5

gmBB~2EF

kT/4

Raw data, low fields

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-1

-0.5

0

0.5

1

dM/dn vs. B @ Density: n=8x1010[cm-2]

Magnetic Field [T]

dM/d

n [m

B]

1.72.74.68.0

Compare with single spins ∂M/∂n=mBtanh(b) , b=gmBB/2T

/ at

1

𝑏∗ ≈ 0 .25

/ at

/ at and

The same characteristic magnetic field

Interactions/

n

n

No interactions

Interactions

d/dn(n), expectations

𝜒

𝜒=𝑛𝜇𝐵

𝑇

d/dn(n), T=1.7-13K

0 2 4 6 8 10-0.5

0

0.5

1

1.5

2

2.5

3

3.5

n [1011 cm-2]

/

n [m

B/T

]

1.7K1.8K2K2.2K2.4K2.7K2.9K3.1K3.3K3.5K3.8K4K4.2K4.6K5.1K5.7K6.9K8K9.2K13.1K

d/dn(n), T=0.6-4K

vs. Temperature/ n

vs. Temperature/ n

/ vs. Temperature and density

Position of the maximum of goes to as

(n), T=1.7-13K

0

( ) ''

n

n dnn

Non-renormalized Pauli susceptibility at

Magnetic moment at B=2T

0

( ) ''

n

n dnnmm

Comparison with Transport Measurements

Main observations

• is nonlinear at surprisingly low characteristic magneticField

• Strong, faster than 1/ divergence• Density at which is maximal related to

the metal-insulator transition

Possible scenario: few electron droplets

• Being created as the density increases• Melted with density and temperature• Typical number spin of a droplet /

Droplet scenario vs theory

• Fermi-liquid expectations:

00

0

, 0.51

FF

Spontaneous large spin droplets in disordered metal

Diffusion enhanced interactions in quantum dots

Mean Field treatment: Andreev, Kamenev (1998)Numerics: Shepelyansky (2001)

Narozhny, B. N. and Aleiner, I. L. and Larkin, A. I. (2000)

0

1/ 2 11

SF

Conclusion:

In the Insulating state of the correlated 2D electron system: spontaneous formation of spin droplets with a large spin S2.

The low field spin susceptibility is strongly temperature dependent (1/T 2) even at high densities,

The spin droplets are detected up to densities well in metallic phase, coexisting with electron liquid

/ changes sign as density or temperature increases. For T 0 this happens right at n=nc

Problems : temperature is unexpected Spin droplets should lead to saturation of dephasing time Role of valley degeneracy is unclear

Problem

O. Prus, Y. Yaish, M. Reznikov, U. Sivan, and V. Pudalov, PRB 2003:

Assumption: at large density the susceptibility is the renormalized Pauli one

0

0( ) ( ) ''

n

n

mm n m dnn

n

This assumption happened to be wrong!

Old results (Prus et al, 2003)

0 2 4 6 8 100.0

0.5

1.0

1.5

2.0

M [1

011 m B

cm-2]

n [1011 cm-2]

T=0.2, 0.8, 2.5, 4.2K B=9.0T 6.0T 4.0T 2.0T 1.0T 0.7T 0.1T full polarization delocalization density

Field dependence of the magnetic moment

0 0.5 1 1.5 2 2.50

1

2

3

4

5

6

7

8 x 1010

B [T]

M/m

b

5.0e+010 [cm-2]8.0e+010 [cm-2]1.6e+011 [cm-2]3.5e+011 [cm-2]8.5e+011 [cm-2]

In-plane magnetoresistance

A. Shashkin et al. PLR, 2001 Fleury, Weintal, 2010.

Raw data

Susceptibility in at B=2T

d/dn(n), Holland sample

0 2 4 6 8 10 12

0

0.5

1

1.5

2

2.5

n [1011 cm-2]

/

n [m

B/T

]

1.8K2.1K2.7K4K4.5K5.8K7.4K8K8K

Stoner Ferromagnetic Instability

Stoner (1947)

2

01B

F

m

Diffusive metal: grows when T

Finkelstein (1983)

0

1 U

For a short range repulsive interaction

Diffusion enhanced interactions in quantum dots

Mean Field treatment: Andreev, Kamenev (1998)Numerics: Shepelyansky (2001)

Clean system

Very small energy difference!

antiferromagnetic

ferromagnetic

A. Finkelstein (1983), Castellani at al.,(1984)Shekhter, A. and Finkel'stein, A. M (2005)

• Higher densities – thermal potential singularities

• Very low density Wigner Crystal rs~37

B. Tanatar and D.C. Ceperley (1989)

Real system

S=0

• Localized electrons Antiferromagnetic coupling

Bhatt and Lee (1982)

Real system

S=0

• Localized electrons Antiferromagnetic coupling

Bhatt and Lee (1982)

Real system

S=0

• Localized electrons Antiferromagnetic coupling

Bhatt and Lee (1982)

• Itinerant electronsDisorder enhances exchange interactions spontaneous formation of finite spin droplets

Andreev, A. V. & Kamenev, A. (1998)Kurland, I. L. and Aleiner, I. L. and Altshuler, B. L. (2000)