Surface Acoustic Wave Propagation in Low density 2DES’s near the Metal-Insulator Transition
2.1 The 2D Metal-Insulator Transition
2.1.2 Theoretical Expectations
The apparent metal-insulator transition came as a surprise. The standard picture (somewhat controversial) is that a “metallic” state is not expected to exist in two dimensions, at least for non-interacting electrons.
Non-interacting picture
For high enough 2DES densities, the electron-electron interactions are relatively weak compared to the kinetic energy. This is due to the fact that the Fermi energy scales linearly with density,EF ∝n, while for the Coulomb interaction EC ∝ n. We first examine this high-density limit, ignoring the effect of electron-electron interactions.
Classical Drude conductivity
Within the Drude picture the 2DES resistivity is given by
CL 2
m
m ρ ne
= τ ,
1 1 1
m i
τ =τ +τ
ie
τ ,
where 1/τm is the momentum relaxation rate (the rate at which the electron undergoes a collision that alters its initial momentum), 1/τi is the impurity scattering rate, and 1/τie is the inelastic scattering rate. As the temperature is reduced, the inelastic relaxation rate vanishes and the resistivity should become a constant as T → 0, where ρ0
is commonly referred to as the residual resistivity.
2
0 m ne/ i
ρ =
Weak localization
Next, including quantum corrections, for a non-interacting 2DES and a small amount of disorder the electron gas is expected to be weakly localized. More precisely, weak localization occurs when , where lm is the mean free path and lφ is the phase- relaxation length. The mean free path is the distance an electron travels between collisions which alter its momentum. The phase-relaxation length is the distance an electron travels before its phase is destroyed due to inelastic collisions (typically due to phonon or electron-electron scattering – see [6] for further discussion of lm and lφ). In this regime, there is a small correction to the conductivity that one would have obtained by simple application of Ohm’s law:
lφ lm
2 2
ln( / )
CL m
e l l
h φ
σ σ
= − π , (1)
where σCL is the classical prediction for the conductivity. This reduction in conductivity from the classical value is due the fact that it is more probable for phase-coherent electrons to backscatter. Performing a sum of the scattering amplitudes over all backscattering paths and the time-reversed versions of those paths leads to a factor of two increase in the total backscattering probability over the classical, non-coherent version of this sum, which would sum over the scattering probabilities, not the amplitudes.
Lowering the temperature tends to lengthen the phase-relaxation length. At low enough temperatures, electron-electron scattering will dominate over the effect of phonons. Electron-electron scattering increases as the temperature is raised and states kBT above and below the Fermi level are filled and emptied. Thus, the weak localization
contribution causes a decrease in the conductivity as the temperature is lowered, giving rise to insulating behavior.
Electron-electron interactions
There is another correction to the temperature dependence of the conductivity due to electron-electron interactions. This term looks similar in form to the weak localization correction, having a logarithmic temperature dependence [7]:
[ ]
2
ln ( F / )( /m B )
e v l k T
δσ h
≈ −π .
Thus, adding weak interactions strengthens the insulating temperature dependence of the weakly-localized state.
Strong localization
For large enough disorder such that , the 2DES is strongly localized. The conductance in this regime is of order or less than e2/h. Conduction occurs via variable range hopping from localized site to site. Efros and Shklovskii [8] argue that the temperature dependence of the conductivity is given by , where p
= 1/2.
m F 1 l k ∼
( )
T 1 exp( / p)σ∼ − −α T
Scaling theory
The scaling theory of localization [9] predicts how the conductance G of a square sample of size L2 scales with system size at zero temperature. Based on various analytical arguments, the theory says that there is a scaling parameter β which is a function of only the dimensionless conductanceg≡G e h/( / )2 , where
[
ln( )]
lnln d g
g d L
β = .
The behavior of this scaling function can be determined by examining some limiting cases. For large, finite conductivities the 2DES is weakly localized and depends on sample size like
2 2
( ) ( )m e ln( / )m
L l L l
σ σ h
= − π ,
where L replaces lφ in Eq. 1 in the zero temperature limit since lφ → ∞. Then 1
β∝ g.
As a check, in the limit , the metallic limit, we obtain which gives Ohm’s law. For small conductivities, strong localization requires that the conductance fall exponentially with length:
g→ ∞ β→0
( )L exp( L) σ ∝ −α . Then
ln( /g g0)
β= ,
where g0 is a constant of order unity. For intermediate conductivities, it is argued on physical grounds that the scaling parameter should be a smooth function of ln(g). The resulting prediction for β is shown in Fig. 2.1. For reference, Fig. 2.1 also shows the scaling for conductance in one and three dimensions as well. In general, in order to obtain Ohm’s Law in the large conductivity limit, we require as , where d is the dimensionality.
2
β→ −d g→ ∞
Fig. 2.1. Scaling of conductivity as proposed by Ref. [5], where d is the dimensionality.
For d = 2, at all finite conductivities, β < 0 implies that all states are localized. Thus, the prediction of scaling theory is that no metallic state should exist in two dimensions.
The case of strong interactions
On the other extreme, we can ignore disorder but consider the case of strong electron- electron interactions. Quantum Monte Carlo calculations predict that the two- dimensional electron gas should crystallize at large rs, where is a dimensionless parameter determining the relative importance of interactions, where EC is the Coulomb energy and EF is the Fermi energy. The most recent calculations of Attaccalite et al. [10] predict that the lowest energy ground state for rs > 35 is a Wigner crystal. Any small amount of disorder should then pin this crystal, leading to insulating behavior at low temperatures.
/ ~ 1/ 2
s C F
r =E E n−
Summary
In reality none of the above mentioned limiting cases completely describe actual experimental systems. When both disorder and strong interactions play a strong role the conclusions become less clear. Interpretation of both theory and experiment in the putative MIT regime remain difficult and controversial.