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)
07/11/11
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
07/11/11
-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
Local moments, strong insulator
diso
rder
electron density
Local moments, strong insulator
diso
rder
electron density
Wignercrystal
Wignercrystal
Paramagnetic Fermi liquid, weak insulator Paramagnetic Fermi
liquid, weak insulator
Ferromagnetic Fermi liquid
Tanatar and Ceperley,
Phys. Rev. B 39, 5005 (1989)
Attaccalite et al.
Phys. Rev. Lett. 88, 256601 (2002)
strength of interactions increases strength of interactions increases
strongly disordered sample
clean sample
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Scaling theory of localization: “all electrons are localized in two dimensions
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
07/11/11 SCCS 2008
silicon MOSFETAl
SiO2 p-Si
2D electrons conductance band
valence band
chemical potential
+ _
ener
gy
distance into the sample (perpendicular to the surface)
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Why Si MOSFETs?
• large m*= 0.19 m0
• two valleys
• low average dielectric constant =7.7
As a result, at low electron densities, Coulomb energy strongly exceeds Fermi energy: EC >> EF
rs = EC / EF >10 can easily be reached in clean samples
EC
EF
EF, E
C
electron density
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Scaling theory of localization: “all electrons are localized in two dimensions
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
07/11/11
Strongly disordered Si MOSFET
(Pudalov et al.)
Consistent (more or less) with the one-parameter scaling theory
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S.V. Kravchenko, G.V. Kravchenko, W. Mason, J. Furneaux, V.M. Pudalov, and M. D’Iorio, PRB 1995
Clean sample, much lower electron densities
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Klapwijk’s sample: Pudalov’s sample:
In very clean samples, the transition is practically universal:
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)
(Note: samples from different sources, measured in different labs)
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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
T = 30 mK
Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRL 2001
The effect of the parallel magnetic field:
07/11/11 SCCS 2008
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
(spins aligned)
Magnetic field, by aligning spins, changes metallic R(T) to insulating:
Such a dramatic reaction on parallel magnetic field suggests unusual spin properties!
07/11/11
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
07/11/11 Pisa 2006
How to study magnetic properties of 2D electrons?
07/11/11
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
T = 30 mK
Spins become fully polarized (Okamoto et al., PRL 1999; Vitkalov et al., PRL 2000)
Method 1: magnetoresistance in a parallel magnetic field
Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRL 2001
Bc
Bc
Bc
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Method 2: weak-field Shubnikov-de Haas oscillations
(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
07/11/11 SCCS 2008
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:
2
,
T
BTB
TBTB
2
,
- diffusive: low temperatures, higher disorder (Tt < 1).
- ballistic: low disorder, higher temperatures (Tt > 1).
The exact formula for magnetoconductance (Lee and Ramakrishnan, 1982):
a
a
F
F
0
0
1 In standard Fermi-liquid notations,
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Experimental results (low-disordered Si MOSFETs; “just metallic” regime; ns= 9.14x1010 cm-2):
S. Anissimova et al., Nature Phys. 3, 707 (2007)
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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!