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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) - PowerPoint PPT Presentation
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Sergey Kravchenko
Approaching an (unknown) phase transition in two dimensions
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)
in collaboration with:
~35 ~1 rs
Wigner crystal Strongly correlated liquid Gas
strength of interactions increases
Coulomb energy Fermi energyrs =
Terra incognita
Distorted lattice Short range order Random electrons
Suggested phase diagrams for strongly interacting electrons in two dimensions
strong insulator
diso
rder
electron density
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
clean sample
strongly disordered sample
University of Virginia
E C
EF
EF,
EC
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:
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.
10/10/09 University of Virginia
Al
SiO2 p-Si
2D electrons conductance band
valence band
chemical potential
+ _
ener
gy
distance into the sample (perpendicular to the surface)
Kravchenko, Mason, Bowker, Furneaux, Pudalov, and D’Iorio, PRB 1995
Metal-insulator transition in 2D semiconductors
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 (
Ohm
)
temperature T (K)
(Note: samples from different sources, measured in different labs)
Sarachik and Kravchenko, PNAS 1999;Kravchenko and Klapwijk, PRL 2000
104
105
106
107
108
109
1010
0 1 2 3 4 5
r (W
)
H|| (Tesla)
Shashkin et al., 2000
Si MOSFET
T = 35 mK
MITn
s just above the zero-field MIT
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 1.2T (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!
Spin susceptibility near nc
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)
Magnetoresistance in a parallel magnetic field
Shashkin, Kravchenko, Dolgopolov, and
Klapwijk, PRL 2001
Bc
Bc
Bc
0
1
2
3
4
5
0 2 4 6 8 10
BB c (meV
)
ns (1015 m-2)
nc
0
0.2
0.4
0.8 1.2 1.6 2
BB c (meV
)
ns (1015 m-2)
Extrapolated polarization field, Bc, vanishes at a finite electron density, n
Shashkin, Kravchenko, Dolgopolov, and
Klapwijk, PRL 2001
Spontaneous spin polarization at n?
n
gm as a function of electron densitycalculated using g*m* = ћ2ns / BcB
1
2
3
4
5
0 2 4 6 8 10
gm/g
0mb
ns (1015 m-2)
ns= n
c
(Shashkin et al., PRL 2001)
n
2D electron gas Ohmic contact
SiO2
Si
Gate
Modulated magnetic fieldB + B
Current amplifierVg
+
-
Magnetic measurements without magnetometer
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
-15 A
)
ns (1011 cm-2)
1 fA!!
Raw magnetization data: induced current vs. gate voltaged/dB = - dM/dn
B|| = 5 tesla
Bar-Ilan University
Raw magnetization data: induced current vs. gate voltageIntegral of the previous slide gives M (ns):complete spin polarization
ns (1011 cm-2)
M (1
011 B
/cm
2 )
met
al
insu
lato
r
0
0.5
1
1.5
0 2 4 6
B|| = 5 tesla
at ns=1.5x1011 cm-2
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 datamagnetocapacitance dataintegral of the master curvetransport data
/ 0
ns (1011 cm-2)
nc
insulator
T-dependent regime
g-factor or effective mass?
Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRB (2002)
0
1
2
3
4
0 2 4 6 8 10
m/m
b , g
/g0
ns (1011 cm-2)
g/g0
m/mb
Effective mass vs. g-factor (from the analysis of the transport data in spirit of
Zala, Narozhny, and Aleiner, PRB 2001) :
Another way to measure m*: amplitude of the weak-field Shubnikov-de Haas oscillations
vs. temperature
(Rahimi, Anissimova, Sakr, Kravchenko, and Klapwijk, PRL 2003)
250
300
350
400
0.2 0.25 0.3 0.35 0.4 0.45 0.5
r (W
/squ
are)
B_|_ (tesla)
430 mK
230 mK
42 mK
1000
2000
3000
4000
0 0.2 0.4 0.6 0.8 1r
(W/s
quar
e)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
Comparison of the effective masses determined by two independent experimental methods:
0
1
2
3
4
0 1 2 3 4
m/m
b
ns (1011 cm-2)
50 30 20 15 12r
s
(Shashkin, Rahimi, Anissimova, Kravchenko, Dolgopolov, and
Klapwijk, PRL 2003)
15 11 8 rs
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
S/T ∝ m/ns
Therefore, divergence of the thermopower indicates a divergenceof the effective mass:
m ∝ ns /(ns − nt)
(Dolgopolov and Gold, 2011)
We observe the increase of the effective mass up to m 25mb 5me!!
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)
A divergence of the effective mass has been predicted…
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 experimentEth = 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 solidformation
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.