Transcript No Slide Title

Approaching an (unknown) phase transition
in two dimensions
Sergey Kravchenko
in collaboration with:
A. Mokashi (Northeastern)
S. Li (City College of New York)
A. A. Shashkin (ISSP Chernogolovka)
V. T. Dolgopolov (ISSP Chernogolovka)
T. M. Klapwijk (TU Delft)
M. P. Sarachik (City College of New York)
rs =
Wigner crystal
~35
Coulomb energy
Fermi energy
Strongly correlated liquid
Terra incognita
Gas
~1
rs
strength of interactions increases
Distorted lattice
Short range order
Random electrons
Suggested phase diagrams for strongly interacting
electrons in two dimensions
Attaccalite et al. Phys. Rev. Lett. 88, 256601
(2002)
Tanatar and Ceperley, Phys. Rev. B 39, 5005
(1989)
strong
insulator sample
strongly
disordered
disorder
disorder
strong insulator
Wigner
crystal
Paramagnetic Fermi
liquid, weak insulator
clean sample
electron density
strength of interactions increases
Wigner
crystal
Ferromagnetic Fermi
liquid
Paramagnetic Fermi
liquid, weak insulator
electron density
strength of interactions increases
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:
University of Virginia
Why Si MOSFETs?
It turns out to be a very convenient 2D system to study strongly-interacting regime
because of:
•
Relatively large effective mass (0.19 m0 )
•
Two valleys in the electronic spectrum
•
Low average dielectric constant =7.7
As a result, at low densities, Coulomb energy strongly exceeds Fermi energy: EC >> EF
rs = EC / EF >10 can be easily reached in clean samples.
For comparison, in n-GaAs/AlGaAs heterostructures, this would require 100 times lower
electron densities. Such samples are not yet available.
Al
SiO2
p-Si
energy
2D electrons
conductance band
chemical potential
valence band
+ _
10/10/09distance
University
of Virginia
into the sample
(perpendicular
to the surface)
Metal-insulator transition in 2D semiconductors
Kravchenko, Mason, Bowker,
Furneaux, Pudalov, and
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:
Shashkin et al., 2000
10
10
9
10
Si MOSFET
T = 35 mK
8
r (W)
10
7
10
6
10
5
MIT 10
n just above the zero-field MIT
s
4
10
0
1
2
3
H|| (Tesla)
4
5
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 (spins
properties!
aligned)
Spin susceptibility near nc
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
10
12
B (Tesla)
Spins become fully polarized
(Okamoto et al., PRL 1999;
Vitkalov et al., PRL 2000)
Extrapolated polarization field, Bc,
vanishes at a finite electron density, n
 B (meV)
5
B c
0.2
B c
 B (meV)
4
0.4
3
0
0.8
1.2
15
n (10
s
2
2
-2
Shashkin, Kravchenko,
Dolgopolov, and
Klapwijk, PRL 2001
m )
nn
1
0
1.6
c
0
2
4
15
ns (10
6
-2
8
10
m )
Spontaneous spin polarization at n?
gm as a function of electron density
calculated using g*m* = ћ2ns / BcB
5
0
gm/g m
b
4
3
(Shashkin et al., PRL 2001)
2
n=n
n
s
1
0
2
c
4
6
15
8
-2
n (10 m )
s
10
Magnetic measurements without magnetometer
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
Raw magnetization
Integral of the
data:
previous
induced
slide
current
givesvs.
M gate
(ns): voltage
complete spin polarization
at ns=1.5x1011 cm-2
2
M (10  /cm )
1.5
B|| = 5 tesla
0.5
metal
insulator
11
B
1
0
0
2
n (10
s
4
11
Bar-Ilan University
6
-2
cm )
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
insulator
2
n
c
T-dependent 1
regime
0.5 1 1.5 2 2.5 3 3.5
n (10
s
11
-2
cm )
g-factor or effective mass?
Effective mass vs. g-factor
(from the analysis of the transport data in spirit of
Zala, Narozhny, and Aleiner, PRB 2001) :
4
m/mb , g/g0
3
m/m
b
Shashkin, Kravchenko,
Dolgopolov, and Klapwijk,
PRB (2002)
2
1
g/g
0
0
0
2
4
11
n (10
s
6
-2
cm )
8
10
Another way to measure m*:
amplitude of the weak-field Shubnikov-de Haas oscillations
vs. temperature
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
0.8
1
B (tesla)
_|_
(Rahimi, Anissimova, Sakr, Kravchenko,
and Klapwijk, PRL 2003)
Comparison of the effective masses determined by two
independent experimental methods:
r
s
15
50 30
20 11 15
rs
128
4
m/m
b
3
2
1
0
0
1
2
3
n (1011 cm-2)
s
4
(Shashkin, Rahimi, Anissimova,
Kravchenko, Dolgopolov, and
Klapwijk, PRL 2003)
Thermopower
Thermopower : S = - V / (T)
S = Sd + Sg = T + Ts
V : heat either end of the sample, measure the induced
voltage difference in the shaded region
T : use two thermometers to determine the temperature
gradient
Divergence of thermopower
1/S tends to vanish at nt
Critical behavior of thermopower
(-T/S) ∝ (ns – nt)x
where x = 1.0±0.1
nt=(7.8±0.1)×1010 cm-2
and is independent of the
level of the disorder
In the low-temperature metallic regime, the diffusion thermopower of
strongly interacting 2D electrons is given by the relation
(Dolgopolov and Gold, 2011)
S/T ∝ m/ns
Therefore, divergence of the thermopower indicates a divergence
of the effective mass:
m ∝ ns /(ns − nt)
We observe the increase of the effective mass up to m  25mb  5me!!
A divergence of the effective mass has been
predicted…
i.
using Gutzwiller's theory (Dolgopolov, JETP Lett. 2002)
ii. using an analogy with He3 near the onset of Wigner crystallization
(Spivak and Kivelson, PRB 2004)
iii. solving an extended Hubbard model using dynamical mean-field
theory (Pankov and Dobrosavljevic, PRB 2008)
iv. from a renormalization group analysis for multi-valley 2D systems
(Punnoose and Finkelstein, Science 2005)
v. by Monte-Carlo simulations (Marchi et al., PRB 2009; Fleury and
Waintal, PRB 2010)
What is the nature of the low-density phase?
Transport properties
If the insulating state were due to a
single-particle localization, the
electric field needed to destroy it
would be of order (the most
conservative estimate)
Eth ~ Wb /le ~ 103 – 104 V/m
However, in experiment
Eth = 1 – 10 V/m !
De-pinning of a pinned Wigner
solid?
Differential resistivity, dV/dI
Broadband noise
Transport properties of the insulating phase favor pinned Wigner solid
formation
SUMMARY:
•
In the clean regime, spin susceptibility critically grows upon approaching to
some sample-independent critical point, n, pointing to the existence of a
phase transition.
•
The dramatic increase of the spin susceptibility is due to the divergence of
the effective mass rather than that of the g-factor and, therefore, is not
related to the Stoner instability. It may be a precursor phase or a direct
transition to the long sought-after Wigner solid.
•
However, the existing data, although consistent with the formation of the
Wigner solid, are not enough to reliably confirm its existence.