07/11/11 SCCS 2008 Sergey Kravchenko in collaboration with: PROFOUND EFFECTS OF ELECTRON-ELECTRON CORRELATIONS IN TWO DIMENSIONS A. Punnoose M. P. Sarachik A. A. Shashkin CCNY CCNY simova V. T. Dolgopolov A. M. Finkelstein T. M. Klapwijk ISSP Texas A&M TU D
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07/11/11SCCS 2008 Sergey Kravchenko in collaboration with: PROFOUND EFFECTS OF ELECTRON-ELECTRON CORRELATIONS IN TWO DIMENSIONS A. Punnoose M. P. Sarachik.
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07/11/11 SCCS 2008
Sergey Kravchenko
in collaboration with:
PROFOUND EFFECTS OF ELECTRON-ELECTRON CORRELATIONS IN TWO DIMENSIONS
A. Punnoose M. P. Sarachik A. A. Shashkin CCNY CCNY ISSP
S. Anissimova V. T. Dolgopolov A. M. Finkelstein T. M. KlapwijkNEU ISSP Texas A&M TU Delft
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Why 2D?
Quantum Hall effect (Nobel Prize 1985)
High-Tc superconductors (Nobel Prize 1988)
FQHE (Nobel Prize 1998)
Graphene (Nobel Prize 2010)
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Outline
Scaling theory of localization: “all electrons are localized in 2D”
Samples
What do experiments show?
Magnetic properties of strongly coupled electrons in 2D: ballistic regime (no disorder)
Interplay between disorder and interactions in 2D; flow diagram
Conclusions
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-4
-3
-2
-1
0
1
3D
2D 1D
MIT
ln(G)
d(lnG)/d(lnL) = (G)
One-parameter scaling theory for non-interacting electrons: the origin of the common wisdom “all states are localized in 2D”
Abrahams, Anderson, Licciardello, and Ramakrishnan, PRL 42, 673 (1979)
G ~ Ld-2 exp(-L/Lloc)
metal (dG/dL>0)insulator
insulator
insulator (dG/dL<0)
Ohm’s law in d dimensions
QM interference
L
G = 1/R
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~1 ~35 rs
Gas Strongly correlated liquid Wigner crystal
Insulator ??????? Insulator
strength of interactions increases
Coulomb energyFermi energyrs =
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Suggested phase diagrams for strongly interacting electrons in two dimensions
(Pudalov et al., PRL 2002; Shashkin et al, 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 1
r (W
/sq
uare
)
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
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2D electron gas Ohmic contact
SiO2
Si
Gate
Modulated magnetic fieldB + B
Current amplifierVg
+
-
Method 3: measurements of thermodynamic magnetization
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
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-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
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0 1 2 3 4 5 6 7
d/d
B
1 B
B = 7 tesla
6 tesla
5 tesla
4 tesla
3 tesla
2 tesla
1.5 tesla
ns (1011 cm-2)
d/dB vs. ns in different parallel magnetic fields:
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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
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Summary of the results obtained by four independent methods (including transport)
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Spin susceptibility exhibits critical behavior near the
sample-independent critical density n : ~ 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
T-dependent regime
Are we approaching a phase transition?
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diso
rder
electron density
Anderson insulator
paramagnetic Fermi-liquidWigner crystal?
Liquid ferromagnet?
Disorder increases at low density and we enter “Punnoose-
Finkelstein regime”
Density-independent disorder
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g-factor or effective mass?
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Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRB 66, 073303 (2002)
0
1
2
3
4
0 2 4 6 8 10
m/m
b ,
g/
g 0
ns (1011 cm-2)
g/g0
m/mb
Effective mass vs. g-factor
Not the Stoner scenario! Wigner crystal? Maybe, but evidence is insufficient
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Effective mass as a function of rs-2 in Si(111) and Si(100)
Si (111)
Si (100)
Shashkin, Kapustin, Deviatov, Dolgopolov, and Kvon, PRB (2007)
Si(111): peak mobility 2.5x103 cm2/Vs
Si(100): peak mobility 3x104 cm2/Vs
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Scaling theory of localization: “all electrons are localized in 2D”
Samples
What do experiments show?
Magnetic properties of strongly coupled electrons in 2D: ballistic regime (no disorder)
Interplay between disorder and interactions in 2D; flow diagram
Conclusions
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Corrections to conductivity due to electron-electron interactions in the diffusive regime (T < 1)
always insulating behavior
However, later this prediction was shown to be incorrect
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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
0
02
2 1ln131ln
2 F
FT
e
Insulating behavior when interactions are weak Metallic behavior when interactions are strongEffective strength of interactions grows as the temperature decreases
Altshuler-Aronov-Lee’s result Finkelstein’s & Castellani-
DiCastro-Lee-Ma’s term
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Punnoose and Finkelstein, Science310, 289 (2005)
interactions
diso
rder
metallic phase stabilized by e-e interaction
disorder takes over
QCP
Recent development: two-loop RG theory
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Low-field magnetoconductance in the diffusive regime yields strength of electron-electron interactions
1
2
Tk
Bg
B
B
22
22
2
1091.0
4,
T
B
k
g
h
eTB
B
B
Experimental testFirst, one needs to ensure that the system is in the diffusive regime (T< 1).
One can distinguish between diffusive and ballistic regimes by studying magnetoconductance:
Temperature dependences of the resistance (a) and strength of interactions (b)
This is the first time effective strength of interactions has been seen to depend on T
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Experimental disorder-interaction flow diagram of the 2D electron liquid
S. Anissimova et al., Nature Phys. 3, 707 (2007)
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Experimental vs. theoretical flow diagram(qualitative comparison b/c the 2-loop theory was developed for multi-valley systems)
S. Anissimova et al., Nature Phys. 3, 707 (2007)
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Quantitative predictions of the one-loop RG for 2-valley systems(Punnoose and Finkelstein, Phys. Rev. Lett. 2002)
Solutions of the RG-equations for r << h/e2:a series of non-monotonic curves r(T). After rescaling, the solutions are described by a single universal curve:
max
max max
ρ(T) = ρ R(η)
η = ρ ln(T /T)
r(T
)
(T)
rmax ln(T/Tmax)
Tmax
rmax
2 = 0.45
For a 2-valley system (like Si MOSFET),
metallic r(T) sets in when 2 > 0.45
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Resistance and interactions vs. T
Note that the metallic behavior sets in when 2 ~ 0.45, exactly as predicted by the RG theory
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Comparison between theory (lines) and experiment (symbols)(no adjustable parameters used!)
S. Anissimova et al., Nature Phys. 3, 707 (2007)
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g-factor grows as T decreases
2.5
3
3.5
4
4.5
0 1 2 3 4
g*
T (K)
ns = 9.9 x 1010 cm-2
“ballistic” value
)1(2* g
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SUMMARY:
Strong interactions in clean two-dimensional systems lead to strong increase and possible divergence of the spin susceptibility: the behavior characteristic of a phase transition
Disorder-interactions flow diagram of the metal-insulator transition clearly reveals a quantum critical point: i.e., there exists a metallic state and a metal-insulator transition in 2D, contrary to the 20-years old paradigm!