Metalinsulator transition MIT in 2D electron gas

1
Metal-insulator transition (MIT) in 2D electron
gas

• Juho Luomahaara 18.10.2007
• Materiaali fysiikka II

2
Introduction

• In 2D electron systems, the electrons are
  confined to move in a plane in the presence of a
  random potential.
• 1D The carriers always strongly localized 3D
  the electronic states either localized or
  extended.
• In 2D the scaling theory of localization
  (Abrahams 1979) predicts insulating behavior
  with decreasing temperature the resistance grows
  (for non-interacting electrons).
• Weak interaction increases the localization even
  further .
• In the strong limit Wigner crystal.
• Therefore 2D systems were not expected to be
  conducting in either limit weak or very strong
  interactions between carriers.
• Experiments performed in the early 1980s on
  different 2D systems confirmed these predictions.
• However, Finkelstein (1984) and Castellani
  (1984) for weak disorder and sufficiently strong
  interactions, a 2D system scales toward a state
  with finite nonzero conductivity

3
Introduction

• Recent progress in semiconductor technology has
  enabled the fabrication of high quality
  two-dimensional samples with very low randomness
  gt very low carrier densities. gt strongly
  correlated electron systems
• Experiments on low-disordered silicon samples
  (Kravchenko et al 1994, 1995, 1996) demonstrated
  dramatic differences between the behaviour of
  strongly interacting systems as compared with
  weakly interacting systems with increasing
  electron density, one can cross from the regime
  where the resistance diverges with decreasing
  temperature (insulating behaviour) to a regime
  where the resistance decreases strongly with
  decreasing T (metallic behaviour).
• Moreover, it was found that in the strongly
  interacting regime, an external magnetic field
  strong enough to polarize the electrons spins,
  induces a giant, positive in-plane
  magnetoresistance and completely suppresses the
  metallic behaviour.

4
Definitions

• The strength of the interactions between
  electrons is usually characterized by the
  dimensionless WignerSeitz radius (in 2D)
• Weak interaction limit rs ltlt 1, strong
  interaction limit rs gtgt 1
• Wigner crystal at rs 35 (in 2D)
• For strong interaction between electrons EC gtgt EF
• Different definitions for MIT
• Derivative method gt metal, gt
  insulator
• Temperature-independent localization length L(ns)
  describes electrons localization on insulating
  side exhibiting divergent behavior near MIT

5
Experimental results in zero magnetic field

• The first experiments that demonstrated the
  unusual temperature dependence of the resistivity
  (Kravchenko et al 1994, 1995, 1996) were
  performed on low-disordered silicon
  metal-oxidesemiconductor field-effect transistors
  (MOSFETs).
• High quality of the samples Mobilities higher
  than before reaching more than 4104 cm2V-1s-1.
  Electron densities below 1011 cm-2.
• EC/EF gt 10 gt Coulomb energy EC is the main
  parameter.
• Metallic (insulating) behaviour with increasing
  (decreasing) electron density

6
Experimental results in zero magnetic field

• In the middle a flat, temperature-independent
  separatrix curve corresponding critical electron
  density nc
• Resistivities of the order h/e2
• The resistivity can be scaled as a function of
  T/T0 with T0 depending only on ns
• Two separate branches (reflection symmetry), the
  upper one for the insulating side of the
  transition and the lower one for the metallic
  side. T 0.2 K 2 K, ns nc -2.5 1010 cm-2
  nc2.5 1010 cm-2
• T0 ? ns ncß with the average power ß
  1.600.1 for the insulating side and 1.620.1 for
  the metallic side of the transition
• Scaling seems to be universal (different samples,
  different groups).

7
Experimental results in zero magnetic field

• For ultra-clean samples behaviour very similar
• The separatrix in ultra-clean samples represents
  the upper limit of the resistivity for which
  metallic behaviour (as characterized by d?/dT gt
  0) can exist metallic ?(T) has never been
  observed in any two-dimensional samples at
  resistivities above3h/e2. A power law of
  mobility.
• In disordered samples resistivity is very
  different. ?(ns) isotherms apparently cross at
  some electron density, the temperature dependence
  of the resistivity does not resemble the critical
  behaviour seen in low-disordered samples. More
  importantly, nc does not coincide with the
  critical density determined by other methods
  (Localization length).
• The transition is not universal in more
  disordered samples and is presumably due to
  Anderson localization, which is strong enough to
  overpower the metallic behaviour at low densities.

8
Experimental results in zero magnetic field

• The strength of the disorder is usually
  characterized by the maximum carrier mobility,
  µmax. In general, the higher the maximum mobility
  (i.e. the lower the disorder), the lower the
  carrier density at which the localization
  transition occurs.
• A metalinsulator transition similar to that seen
  in clean silicon MOSFETs has also been observed
  in p-type SiGe and AlAs heterostructures
• Therefore, ?(T ) curves are nearly universal in
  the vicinity of the metalinsulator transition,
  but only in samples with very weak disorder
  potential (different methods, different samples).
• Since one of the methods is independent of
  temperature, this equivalence supports the
  existence of a true T 0 MIT in low-disordered
  samples in zero magnetic field.

9
The effect of the magnetic field

• In ordinary metals, the application of a parallel
  magnetic field (B) does not lead to any dramatic
  changes in the transport properties if the
  thickness of the two-dimensional electron system
  is small compared to the magnetic length, the
  parallel field couples largely to the electrons
  spins while the orbital (motion) effects are
  suppressed.
• However, Dolgopolov et al (1992) observed a
  dramatic suppression of the conductivity in
  dilute Si MOSFETs by a parallel in-plane magnetic
  field B.

10
The effect of the magnetic field

• The resistivity increases sharply as the magnetic
  field is raised and saturates and remains
  approximately constant up to the highest
  measuring field, B 12 T.
• Bsat Bsat(ns), independent of the orientations
  of the magnetic field and measuring current
  (isotropic systems) gt the giant
  magnetoresistance is due to coupling of the
  magnetic field to the electrons spins
• Okamoto et al (1999) and Vitkalov et al (2000,
  2001a) The magnetic field Bsat is equal to that
  required to fully polarize the electrons spins.
• An important effect of a parallel field is that
  it causes the zero-field two-dimensional metal to
  become an insulator.
• The effect of the field is negligible at
  temperatures above T 2 K.
• The spin-polarized and unpolarized states behave
  very differently.
• Magnetoresistance is qualitatively the same for
  carrier densities above and below the zero-field
  critical density nc. gt The physical mechanism
  that gives rise to the magnetoresistance is the
  same for metallic and insulating side.
• Attempts to obtain a quantitative description of
  the magnetoresistance as a function of the
  carrier density and temperature over the entire
  field range B have been unsuccesful.
• Fermi liquid theory predicts an increase in spin
  suceptibility ? when ns is decreased, a fact that
  has been confirmed by experiments.

11
Problems

• Metallic behavior is displayed down to the lowest
  temperatures under conditions in which 2D systems
  are expected to show insulating behavior because
  of localization due to disorder (Anderson
  localization).
• The application of a magnetic field at an
  arbitrary angle to the plane of the
  two-dimensional electron liquid suppresses the
  metallic behavior and restores localization and
  other normal properties.

12
Theories

• Very little theory has been developed for
  strongly interacting systems for which rs is
  large but below the expected Wigner
  crystallization.
• Several candidates (i) a Wigner crystal
  characterized by spatial and spin ordering
  (Wigner 1934), (ii) an itinerant ferromagnet with
  spontaneous spin ordering (Stoner 1946) and (iii)
  a paramagnetic Fermi liquid (Landau 1957)
• Recent detailed numerical simulations
  (Attaccalite et al 2002) have predicted that in
  the range of the interaction parameter 25 lt rs lt
  35 prior to the crystallization, the ground state
  of the system becomes an itinerant ferromagnet.
• As discussed earlier, there are experimental
  indications that a spontaneous spin polarization
  may occur at a finite electron density in silicon
  MOSFETs.

13
Theories based on Fermi liquid

• Finkelstein (1983, 1984) and Castellani et al
  (1984) combined effects of disorder and
  interactions were by perturbative renormalization
  group methods.
• It was found that as the temperature is
  decreased, the resistivity increases and then
  decreases at lower temperatures, suggesting that
  the system is approaching a low temperature
  metallic state. An external magnetic field, via
  Zeeman splitting, drives the system back to the
  insulating state. These predictions of the theory
  are in qualitative agreement with experiments.
• However, an interaction parameter scales to
  infinitely large values before zero temperature
  is reached, and the theory becomes uncontrolled
• Ballistic regime, far from the transition (ns gtgt
  nc) Zala et al (2001) has shown the temperature
  dependence of the conductivity in the ballistic
  regime originates from coherent scattering of
  electrons by Friedel oscillations. The phase of
  the Friedel oscillation is such that the wave
  scattered from the impurity interferes
  constructively with the wave scattered from the
  oscillation, leading to a correction linear
  relation with respect to T.
• Diffusive regime, a relatively narrow range of
  electron densities near the metalinsulator
  transition Punnoose and Finkelstein (2002) have
  convincingly demonstrated that in this region,
  the temperature dependence of the resistivity can
  be understood within the renormalization group
  theory that considers the interplay of the e-e
  interaction and disorder.

14
Theories based on Fermi liquid

• In both cases, ballistic and diffusive regime, an
  external magnetic field quenches the delocalizing
  effect of interactions by aligning the spins,
  causing a giant positive magnetoresistance.

15
Conclusions

• The metalinsulator transition is not yet
  understood theoretically.
• Various descriptions have been proposed, ranging
  from the melting of a Wigner solid from the
  insulating side to the formation of one from the
  metallic side from superconductivity to quantum
  percolation from a semiclassical one-electron
  description with no metal-insulator transition to
  a non-Fermi-liquid scenario. While each of these
  is capable of explaining one or another part of
  the set of experimental observations, none of
  them provides a comprehensive picture
• A central question that must be answered by
  experiment is whether there is a true
  metal-insulator transition at ns nc and a
  metallic phase at intermediate densities between
  nc and the higher densities where localization is
  known to prevail.
• More measurements need to be performed
  tunneling, magnetization, specific heat, electron
  spin resonance, nuclear magnetic resonance
  Difficult.
• The main theoretical issue is the description of
  the 2D electron (or hole) system in the
  neighborhood of the critical density.
• Experiments require a theory of the unusual
  metallic phase. (The enermous parallel-field
  magnetoresistance)

16
References

• Metalinsulator transition in two-dimensional
  electron systems S V Kravchenko and M P
  Sarachik Rep. Prog. Phys. 67 (2004)
• Colloquium Metallic behavior and related
  phenomena in two dimensions Abrahams,
  Kravchenko, Sarachik REVIEWS OF MODERN PHYSICS,
  VOLUME 73, APRIL 2001