Transcript No Slide Title

AMAZING PROPERTIES OF STRONGLY CORRELATED
ELECTRONS IN TWO DIMENSIONS
Sergey Kravchenko
in collaboration with:
S. Anissimova
NEU
A. Punnoose
CCNY
07/11/11
V. T. Dolgopolov A. M. Finkelstein T. M. Klapwijk
ISSP
Texas A&M
TU Delft
M. P. Sarachik
CCNY
A. A. Shashkin
ISSP
SCCS 2008
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:
Conduction band
Conduction band
EF
Valence 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
Electrons can penetrate infinitely far
(with some scattering)
High disorder
Electrons are localized at a certain
distance, called “localization length”, Lloc
Llo
c
“… 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.
But electrons do interact via Coulomb forces!
rs =
Gas
Coulomb energy
Fermi energy
Strongly correlated liquid
~1
Insulator
Wigner crystal
~35
Terra incognita
strength of interactions increases
rs
Insulator
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:
EF
EF, EC
EC
University of Virginia
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
silicon MOSFET
Al
SiO2
p-Si
energy
2D electrons
conductance band
chemical potential
valence band
+ _
07/11/11distance
SCCS 2008
into the sample (perpendicular
to the surface)
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
EF
EF, EC
EC
SCCS 2008
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
Clean sample, much lower electron densities
S.V. Kravchenko, G.V.
Kravchenko, W. Mason, J.
Furneaux, V.M. Pudalov, and
M. D’Iorio, PRB 1995
In very clean samples, the transition is practically universal:
Sarachik and Kravchenko,
PNAS 1999;
Kravchenko and Klapwijk,
PRL 2000
6
resistivity r (Ohm)
10
5
10
11
4
(Note: samples from
different sources,
measured in different labs)
3
10
-2
0.86x10 cm
0.88
0.90
0.93
0.95
0.99
1.10
10
0
0.5
1
temperature T (K)
1.5
2
The effect of the parallel magnetic field:
5
10
15
1.01x10
-2
m
15
T = 30 mK
r (Ohm)
1.20x10
4
10
15
1.68x10
15
2.40x10
15
3.18x10
3
10
0
2
4
6
8
B (Tesla)
10
12
Shashkin, Kravchenko,
Dolgopolov, and
Klapwijk, PRL 2001
Magnetic field, by aligning spins, changes metallic R(T) to insulating:
Shashkin et al., 2000
6
6
r (W)
10
10
0.765
0.780
0.795
0.810
0.825
1.095
1.125
1.155
1.185
1.215
5
10
5
4
10
10
10
4
0
0.3
0.6
T (K)
0.9
1.2
0
0.3
0.6
0.9
1.2
T (K)
B =a 0
B>B
Such
dramatic reaction on parallel
magnetic
sat
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
Method 1: magnetoresistance in a parallel magnetic field
5
10
15
1.01x10
-2
m
15
1.20x10
T = 30 mK
r (Ohm)
Bc
4
10
15
1.68x10
Bc
Bc 2.40x1015
Shashkin, Kravchenko,
Dolgopolov, and
Klapwijk, PRL 2001
15
3.18x10
3
10
0
2
4
6
8
B (Tesla)
10
12
Spins become fully polarized
(Okamoto et al., PRL 1999;
Vitkalov et al., PRL 2000)
Method 2: weak-field Shubnikov-de Haas oscillations
high density
low density
400
4000
350
430 mK
230 mK
42 mK
300
r (W/square)
r (W/square)
 =14
 =10
3000
3100
132 mK
=6
3000
2000
2900
2800
82 mK
0.3
250
0.2
T = 42 mK
0.4
42 mK
0.5
0.6
1000
0.25
0.3
0.35
B (tesla)
_|_
0.4
0.45
0.5
0
0.2
0.4
0.6
B (tesla)
_|_
(Pudalov et al., PRL 2002; Shashkin et al, PRL 2003)
0.8
1
Method 3: measurements of thermodynamic magnetization
suggested by B. I. Halperin (1998); first implemented by O. Prus, M. Reznikov, U. Sivan et al. (2002)
1010 Ohm
+
Gate
Vg
Current amplifier
SiO2
Si
2D electron gas
Modulated magnetic field
B + B
Ohmic contact
i ~ d/dB = - dM/dns
Raw magnetization data:
current vs. gate voltage
d/dBinduced
= - dM/dn
1
2
0.5
-15
i (10 A)
B
1
d/dB ( )
1 fA!!
0
0
-1
-0.5
B|| = 5 tesla
-2
-1
0
1
2
3
4
11
n (10
s
5
-2
cm )
6
7
Spin susceptibility exhibits critical behavior near the
sample-independent critical density n :
 ~ ns/(ns – n)
7
6
magnetization data
magnetocapacitance data
integral of the master curve
transport data
/
0
5
4
3
2
n
c
1
0.5 1 1.5 2 2.5 3 3.5
n (10
11
-2
cm )
s approaching a phase transition?
Are we
g-factor or effective mass?
Effective mass vs. g-factor
4
m/mb , g/g0
3
m/m
b
2
Shashkin, Kravchenko,
Dolgopolov, and Klapwijk,
PRB 66, 073303 (2002)
1
g/g
0
0
0
4
2
11
n (10
s
6
-2
8
10
cm )
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