Spin-droplet state of an interacting 2D electron system
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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
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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
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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
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Phase diagramAttaccalite et al. (2001)
First order transition at rs~20: Senatore et al. (2001)
rs~26
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Clean system
• Very low density Wigner Crystal rs~37
B. Tanatar and D.C. Ceperley (1989)
ferromagnetic
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Clean system
Very small energy difference!
antiferromagnetic
ferromagnetic
• Very low density Wigner Crystal rs~37
B. Tanatar and D.C. Ceperley (1989)
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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
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V. Pudalov at al, (2001)
Metal-Insulator Transition in a Silicon Inversion Layer
m
gmBB
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In-plane magnetoresistance
S. Vitkalov et al. PRL 2001A. Shashkin et al. PLR, 2001
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In-plane magnetoresistance
A. Shashkin et al. PLR, 2001
Possible FM transition ??
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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]
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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
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Finite thickness contributions to at
Diamagnetic contribution
Capacitance contribution
change
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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!
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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
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B (T)
/
/ at n=1.5
gmBB~2EF
kT/4
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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
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/ at
1
𝑏∗ ≈ 0 .25
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/ at
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/ at and
The same characteristic magnetic field
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Interactions/
n
n
No interactions
Interactions
d/dn(n), expectations
𝜒
𝜒=𝑛𝜇𝐵
𝑇
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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
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d/dn(n), T=0.6-4K
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vs. Temperature/ n
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vs. Temperature/ n
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/ vs. Temperature and density
Position of the maximum of goes to as
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(n), T=1.7-13K
0
( ) ''
n
n dnn
Non-renormalized Pauli susceptibility at
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Magnetic moment at B=2T
0
( ) ''
n
n dnnmm
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Comparison with Transport Measurements
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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 /
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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
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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
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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!
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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
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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]
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In-plane magnetoresistance
A. Shashkin et al. PLR, 2001 Fleury, Weintal, 2010.
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Raw data
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Susceptibility in at B=2T
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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
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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)
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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)
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Real system
S=0
• Localized electrons Antiferromagnetic coupling
Bhatt and Lee (1982)
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Real system
S=0
• Localized electrons Antiferromagnetic coupling
Bhatt and Lee (1982)
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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)
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