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07/11/11SCCS 2008 Sergey Kravchenko in collaboration with: AMAZING PROPERTIES OF STRONGLY CORRELATED...
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Transcript of 07/11/11SCCS 2008 Sergey Kravchenko in collaboration with: AMAZING PROPERTIES OF STRONGLY CORRELATED...
07/11/11 SCCS 2008
Sergey Kravchenko
in collaboration with:
AMAZING PROPERTIES OF STRONGLY CORRELATED ELECTRONS 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
Outline
Scaling theory of localization: “all electrons are localized in 2D”
Samples
What do experiments show?
Magnetic properties of strongly correlated electrons in 2D
Conclusions
Band theory of metals:
E F
Conduction band
Valence band
Conduction band
Insulator Metal
But it turns out that even if the Fermi level lies in the conduction band,the system may be insulating.
Localization by disorder (Anderson localization)
Low disorder High disorder
Electrons can penetrate infinitely far(with some scattering)
Electrons are localized at a certain distance, called “localization length”, Lloc
Lloc
“… very few believed in [localization] at the time, and even fewer saw its importance... It has yet to receive adequate mathematical treatment, and one has to resort to the indignity of numerical simulations to settle even the simplest questions about it."
P.W. Anderson, Nobel Lecture, 1977
In 1979, a powerful theory was created by the “Gang of Four” (Abrahams, Anderson, Licciardello, and Ramakrishnan), according to
which, there is no conductivity in 2D at low temperatures.
This became one of the most influential paradigms in modern condensed matter physics.
However, this prediction is valid for non-interacting electrons only.
~1 ~35 rs
Gas Strongly correlated liquid Wigner crystal
Insulator Insulator
strength of interactions increases
Coulomb energy Fermi energyrs =
Terra incognita
But electrons do interact via Coulomb forces!
University of Virginia
EC
EF
EF, E
C
electron density
In 2D, the kinetic (Fermi) energy is proportional to the electron density:
EF = (h2/m) Ns
while the potential (Coulomb) energy is proportional to Ns1/2:
EC = (e2/ε) Ns1/2
Therefore, the relative strength of interactions increases as the density decreases:
Scaling theory of localization: “all electrons are localized in two dimensions
Samples
What do experiments show?
Magnetic properties of strongly correlated electrons in 2D
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)
SCCS 2008
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
Scaling theory of localization: “all electrons are localized in two dimensions
Samples
What do experiments show?
Magnetic properties of strongly correlated electrons in 2D
Conclusions
Strongly disordered Si MOSFET
(Pudalov et al.)
Consistent (more or less) with the one-parameter scaling theory
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
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)
tem perature T (K )
(Note: samples from different sources, measured in different labs)
Sarachik and Kravchenko, PNAS 1999;Kravchenko and Klapwijk, PRL 2000
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:
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
Magnetic field, by aligning spins, changes metallic R(T) to insulating:
Such a dramatic reaction on parallel magnetic field suggests unusual spin properties!
Scaling theory of localization: “all electrons are localized in 2D”
Samples
What do experiments show?
Magnetic properties of strongly correlated electrons in 2D
Conclusions
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
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
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
-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
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
Are we approaching a phase transition?
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
SUMMARY:
(i) There exists a metallic state in 2D, contrary to the 30-years old paradigm!
(ii) 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
(iii) The dramatic increase of the spin susceptibility is caused by the effective mass rather than by the g-factor