Effects of electron-electron interactions in two-dimensional electron systems

1
Effects of electron-electron interactions in
two-dimensional electron systems
A. A. Shashkin

• Institute of Solid State
  Physics RAS

2
In collaboration with

Sveta Anissimova Sergey Kravchenko Vadim
Khrapai Valeri Dolgopolov Teun Klapwijk
3
Why Si MOSFETs?

• EC gtgt EF
• m 0.19m0
• two valleys
• e 7.7
• EC/EF 18
• 
  
  

4
Si (100) MOSFET
ml0.916 me , mt0.19 me
5
silicon MOSFET
Al
SiO2
p-Si
conductance band
2D electrons
chemical potential
energy
valence band
_

distance into the sample (perpendicular to the
surface)
6
Split-gate technique dramatically reduces contact
resistance
high-density region
gate
low-density region
100 nm slit in the gate metallization
ohmic contacts
7
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)
Wigner crystal
Ferromagnetic Fermi liquid
Paramagnetic Fermi liquid
Wigner crystal
Paramagnetic Fermi liquid
strength of interactions increases
strength of interactions increases
8
Metal-insulator transition is seen in clean
strongly-interacting 2D systems
clean sample
(in contrast to strongly disordered systems)
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Whats the underlying physics of the
effect?
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Beating pattern of Shubnikov-de Haas
oscillations in perpendicular B
low density
high density
Near the MIT, spin splitting
cyclotron splitting, or gm/2me1. This
is higher by more than a factor of 5 than the
band value gm/2me0.19.
Kravchenko et al., Solid State Commun. 116, 495
(2000)
11
Beating pattern of Shubnikov-de Haas
oscillations in perpendicular B
Cyclotron gaps at n4, 8, 12, 16 disappear as the
electron density is reduced
Kravchenko et al., Solid State Commun. 116, 495
(2000)
12
Magnetoresistance in a parallel magnetic field
T 30 mK
Bc
Shashkin, Kravchenko, Dolgopolov, and Klapwijk,
PRL 2001
Bc
Bc
Spins become fully polarized (Okamoto et al.,
PRL 1999 Vitkalov et al., PRL 2000)
13
Scaling the magnetoresistance yields Bc(ns)
ns (1015 m-2)
Shashkin, Kravchenko, Dolgopolov, and Klapwijk,
PRL 2001
Theoretical dependence
(Dolgopolov and Gold, JETP Lett. 2000)
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Extrapolated polarization field, Bc, vanishes at
a finite electron density, nc
nc _at_ nc for the MIT
Shashkin, Kravchenko, Dolgopolov, and Klapwijk,
PRL 2001
nc
Spontaneous spin polarization at nc?
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gm as a function of electron densitycalculated
using gm p?2ns / mBBc
Shashkin et al., PRL 2001
nc
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Enhanced gm values obtained at low electron
densities by SdH oscillations and parallel-field
magnetotransport are in agreement with each
other. Therefore, the band tail of localized
electron states is small, which corresponds to
the clean regime. Vanishing Bc at a finite nc _at_
nc indicates a ferromagnetic transition in this
electron system. The fact that nc _at_ nc
indicates that the MIT is driven by interactions.
Consequences
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Local moments in the band tail
Mott and Davis, Electronic Processes in
Non-Crystalline Materials (Clarendon Press,
Oxford, UK, 1971)
Spin magnetization in parallel B
Gold and Dolgopolov, JPCM 14, 7091 (2002)
SdH oscillations correspond to total electron
density
(Fowler et al., PRL 1966)
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In clean samples, the same value nc _at_ 8.1010 cm-2
is obtained
by temperature-independent resistance
(requires extrapolation to T0)
Shashkin, Kravchenko, and Klapwijk, PRL 2001
nc
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and by a vanishing activation energy combined
with a vanishing nonlinearity of I-V
curves (temperature-independent)
Shashkin, Kravchenko, and Klapwijk, PRL 2001
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Comparison to other groups data
Shashkin et al, 2001
Pudalov et al, 2002
Vitkalov, Sarachik et al, 2001
nc
nc _at_ 8.1010 cm-2 is sample-independent
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How to separate g and
m? Which of the two is
responsible for the
renormalization of c?
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Temperature dependence of conductivity
Ds/s -C(ns) kT/EF
Gold and Dolgopolov, PRB 33, 1076 (1986)
Ds/s (1aF0s)/(1F0s) kT/EF g 2/(1 F0s)
Ds/s -AkT
A -(1aF0s)gm/p?2ns -(1aF0s)/mBBc
Zala, Narozhny, and Aleiner, PRB 64, 214204 (2001)
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Indeed, conductivity in the metallic regime is a
linear function of T (not too close to the
transition)
Shashkin, Kravchenko, Dolgopolov, and Klapwijk,
PRB 2002
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1/A µBBc yields g(ns) const.This
conclusion is independent of the value of A and
of how accurately Bc is determined (only
functional forms are important)
µBBc
1/A
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Shashkin, Kravchenko, Dolgopolov, and Klapwijk,
PRB 66, 073303 (2002)
Effective mass vs. g-factor
Spin effects are not pronounced
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Shashkin, Kravchenko, Dolgopolov, and Klapwijk,
PRB 2002
Scattering time
renormalized m
m mb
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Data do not favor either theory
2.46
2.23.1011 cm-2
Data of Pudalov et al., PRL 2003
An anomalous increase of r with temperature
can be described quantitatively by the
interaction effects in the ballistic regime.
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Damping of the weak-field Shubnikov-de Haas
oscillations with temperature
high density
low density
Pudalov et al., PRL 2002 Shashkin et al, PRL 2003
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Oscillations numbers n10 (dots) and n14
(squares). The value of T for the n 10 data is
divided by the factor of 1.4.
ns1.17.1011 cm-2
Shashkin, Rahimi, Anissimova, Kravchenko,
Dolgopolov, and Klapwijk, PRL 2003
The amplitude of the SdH oscillations follows
the calculated curve down to the lowest achieved
temperature the electrons are in a good thermal
contact with the bath.
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Comparison of the effective masses determined by
two independent experimental methods
recalculated ratio Ee-e /EF
Shashkin, Rahimi, Anissimova, Kravchenko,
Dolgopolov, and Klapwijk, PRL 2003
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The effective mass is independent of the degree
of spin polarization
Therefore, the renormalization of m is not
related to spin exchange effects
Electron densities 1.32 , 1.47, 2.07, 2.67.1011
cm-2.
Shashkin, Rahimi, Anissimova, Kravchenko,
Dolgopolov, and Klapwijk, PRL 2003
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Magnetocapacitance spectroscopy
Khrapai, Shashkin, and Dolgopolov, PRB 2003
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Spin gap
Khrapai, Shashkin, and Dolgopolov, PRL 2003
g _at_ 2.6 does not change with filling factor
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Cyclotron gap
me/m-g
Khrapai, Shashkin, and Dolgopolov, PRL 2003
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• CONCLUSIONS
• In strongly interacting 2D electron system in
  silicon,
• spin susceptibility sharply rises with a
  tendency to
• diverge at a sample-independent density nc.
• We find no evidence of increasing g-factor it
  must be
• the effective mass that is responsible for the
  effect.
• In clean samples, nc practically coincides with
  the
• metal-insulator transition point.

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Electrons in GaAs/AlGaAs

• absence of valley splitting (hence larger Fermi
  energy)
• smaller effective mass (again, larger Fermi
  energy)
• larger dielectric constant (weaker Coulomb
  interactions)
• much wider electron density distribution in
  zdirection.

As a result, at a fixed electron density, the
interaction parameter in GaAs/AlGaAs is 10 times
smaller than that in Si MOSFETs. Characteristic
electron densities in GaAs/AlGaAs should be 100
times smaller nslt 109 cm-2. They have not been
reached in experiments yet.
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Achievable densities are well above the expected
critical region
Data of Zhu et al., PRL 2003
38
Valley gap at n1, 3, and 5
Khrapai, Shashkin, and Dolgopolov, PRB 2003
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Difference between our results and those of
Prus, Reznikov, Sivan et al. (PRB 2003)
Only these data points are relevant to the
metallic regime
m
Data of Prus, Reznikov, Sivan et al., PRB 2003
40
Indeed, the (inverse) spin susceptibility has a
Curie form characteristic of local moments
Data of Prus, Reznikov, Sivan et al., PRB 2003
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Analyzing the SdH oscillations at T gt 0.3 K, one
cannot establish which value (either g or m or
both) is responsible for the enhanced c
Data of Pudalov et al., PRL 88, 196404 (2002)
Below 0.3 K, we observed the trend of
simultaneous saturation of the temperature
dependences of r0, drxx, and the dephasing rate,
which is presumably caused by electron
overheating.
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Pudalov et al. have claimed that their data are
inconsistent with ours
Data of Pudalov et al., PRL 2002
(Reply to Comment by Kravchenko, Shashkin and
Dolgopolov, PRL 2002)
We determined the spin susceptibility c, the
effective mass m, and the g factor for mobile
electrons. These quantities increase gradually
with decreasing density.
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How were these three points obtained?
Certainly, T gt 0.3 K is way too high!!
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Oscillations in Si MOSFET at low electron
density Lifshitz-Kosevich formula can hardly be
used to extract m
clearly not ordinary SdH oscillations!!
ns 9.3x1010 cm-2
DIorio, Pudalov, and Semenchinsky, Surf. Sci.
(1994)
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MIT in perpendicular B
Shashkin et al., JETPL 1993 Kravchenko et al.,
PRL 1995