2D-MIT as self-doping of aWigner-Mott insulator
S. Pankov ∗ V. Dobrosavljevic
National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306
Abstract
We consider an interaction-driven scenario for the two-dimensional metal-insulator transition in zero magnetic field (2D-MIT),
based on melting the Wigner crystal through vacancy-interstitial pair formation. We show that the transition from the Wigner-
Mott insulator to a heavy Fermi liquid emerges as an instability to self-doping, resembling conceptually the solid to normal liquid
transition in He3. The resulting physical picture naturally explains many puzzling features of the 2D-MIT.
Key words: Strong correlation; disorder; metal-insulator transition; Hubbard model
PACS: 71.27.+a, 72.15.Rn, 71.30.+h
A series of fascinating experiments, as first performed
some ten years ago by Kravchenko and co-workers [1], have
deeply changed our thinking about the two dimensional
electron gas (2DEG). These and many later experiments
demonstrated convincingly that the 2DEG can exhibit typ-
ical metallic behavior [2] above a well defined critical den-
sity nc. One of the most prominent features of this metallic
phase is an unprecedented resistivity drop, which is found
only in the low density regime n & nc close to the tran-
sition. These findings suggested that a well-defined metal-
insulator transition (MIT) may exist even in two dimen-
sions, in contrast to long held-beliefs based on the theories
for noninteracting disordered electrons.
These experiments are typically performed at such low
electron density where the relative strength of the Coulomb
interaction is so large (rs & 10), that the localization pro-
cesses could conceivably be suppressed by interaction ef-
fects. A diffusion-mode theory describing such interaction
renormalizations at weak disorder has been developed by
Finkelshtein and Punnoose [3], suggesting that sufficiently
strong interactions may indeed stabilize the metallic phase.
However, this theory can provide guidance only within a
narrowdiffusive regime restricted to very low temperatures.
In contrast, the most striking experimental results have
been established in a broad parameter range well outside
this regime, and are most pronounced in the cleanest sam-
ples [2]. In particular, the best established experimental sig-
nature of the transition relies on careful effective mass mea-
surements, which is found to diverges as m∗ ∝ (n− nc)
∗ Corresponding author; email: pankov@magnet.fsu.edu
1 1.05 1.1
-1.002
-0.998
-0.996
Fig. 1. Evolution of the free energy profile W [δ], as the electron
density is increased across the MIT (from top to bottom). The cusp
corresponds to the Wigner solid. The self-doped transition (blue
line) takes place before the instability emerges at half-filling (δ = 0,
orange line) and the Wigner insulating state is replaced by a heavy
Fermi liquid.
while the Lande g∗-factor remains largely unrenormalized.
Such phenomena cannot be understood within a low-energy
diffusion mode theory, which relies on Anderson localiza-
tion to produce an insulating phase.
A fundamental question is thus posed by these experi-
ments: what is the basic mechanism that drives the metal-
insulator transition in these systems? Does one have to rely
on disorder effects at all in the zero-th order approximation,
or can one understand the most important experimental
features by interaction effects alone? In this work, we con-
centrate solely on the effects of strong Coulomb interactions
in the clean limit, and try to establish which experimental
Preprint submitted to Elsevier 31 October 2018
http://arxiv.org/abs/0704.0250v1
features can be explained by entirely ignoring the disorder.
We approach the transition from the insulating side,
starting with the Wigner-Mott insulator, and examine
its melting by quantum fluctuations as density increases.
These are believed to be dominated [4] by the vacancy-
interstitial pair excitations on top of the classical triangu-
lar lattice configuration. Within the Wigner crystal, the
electrons are tightly bound, and the vacancy-interstitial
excitations can be well represented by a mapping to a two-
band Hubbard (i.e. charge-transfer) model, respectively
corresponding to the lattice and the interstitial electrons.
As the density increases, the charge-transfer gap eventu-
ally closes, and a metal-insulator transition assumes the
character of a Mott-metal-insulator transition, leading to
a strongly correlated metallic state on the metallic side.
Our model Hamiltonian reads:
fiσ + ecc
ciσ −
iσciσ + c
iσfiσ) +
fi↓ (1)
where f †, f and c†, c are creation and annihilation opera-
tors for site and interstitial electrons respectively. It is as-
sumed that the inter-cell hopping tij exists only between
interstitial orbitals, while only the site electrons are subject
to onsite Coulomb repulsion U , and the interstitial orbitals
are coupled to the site orbitals via hybridization V . The
local electrostatic potentials for the two bands are denoted
by ef and ec.
Wemodel the strong onsite repulsion by exclusion of dou-
ble ocupancy, using the standard slave-boson mean-field
formalism. The free energy per electron then reads:
W [λ, Z, µ, δ] =
ln (1 + exp (−(ε̃lk − µ)/T ))
(Z − 1) + µ (2)
where εlk are renormalized band energies, λ is the Lagrange
multiplier enforcing the slave-boson constraint, Z is the
quasiparticle weight, µ is the chemical potential. The chem-
ical potential µ is an internal parameter here, arising simi-
larly to λ, from constraining the electron density per unit
area. The self-doping δ measures deviation in the number
of electrons (per elementary cell) from the half filling. In
the classical limit of low electron density δ = 0, but it be-
comes δ 6= 0 at higher density, where quantum effects are
important. This is reminiscent of the liquid solid transition
in He3 [5]. The equations of the state are given by the sad-
dle point of the free energy W [λ, Z, µ, δ].
A peculiarity of this model is that the band energies
are not fixed, but are self-consistently determined through
their dependence on the occupation of the site and inter-
stitial orbitals. It immediately follows that the system is
unstable to the self-doping at the MIT. Indeed, by care-
fully accounting for the electrostatic energy balance due to
such charge transfer, we find that the free energy takes a
lower value (relative to the classical value at δ = 0) on one
of the branches at δ 6= 0 before the transition at half filling
is found (see Fig. 1).
With appropriately chosen model parameters we can ex-
plain the basic experimental results, which otherwise can-
not be captured by the diffusion mode theory. Similarly
to what is seen experimentally, we observe strong renor-
malization of the effective mass near the transition m∗ ∼
(n − nc)
−1. In this strongly correlated regime any extrin-
sic disorder is very effectively screened by interaction ef-
fects [7], providing a plausible scenario for the large resis-
tivity drop [8]. The magneto-resistance data are also natu-
rally explained by our two band model. The experiment [9]
has shown that in high parallel magnetic fields the trans-
port takes place by activated processes, with an activation
gap that vanishes linearly at some density n1 > nc. In the
strong field our model reduces to a trivial model of non-
interacting spinless electrons. For the density n < n1 the
system is a band insulator with the lower band completely
filled. The bands broaden as the density increases, and the
insulating gap closes linearly at some density n1 > nc.
Our static lattice model does not capture the collective
charge density fluctuations – the phonons of the Wigner
crystal. Because the Coulomb interaction is long ranged,
these collective modes are very soft, and play an important
role in renormalizing the model parameters. It is this strong
renormalization of the charge transfer gap ec − ef that
pushes the transition to such low density [6] (rs ≫ 1)), in a
fashion that is conceptually very similar to the formation of
the Coulomb gap [10]. At present, these effects are incorpo-
rated in through the choice of the electrostatic parameters
of our model. In future work, we would like to systemati-
cally incorporate these soft collective modes, the effects of
which are currently included in a semi-phenomenological
fashion. This program can be achieved by a variety of meth-
ods, including extended dynamical mean field approaches
(EDMFT), by exploiting unique properties of the Coulomb
potential, and by employing the techniques recently devel-
oped in the Coulomb glass context [10].
References
[1] S. V. Kravchenko et al., Phys. Rev. B 51, 7038 (1995).
[2] For a review, see S. V. Kravchenko and M. P. Sarachik, Rep.
Prog. Phys. 67, 1 (2004).
[3] A. Punnoose and A. M. Finkelstein, Phys. Rev. Lett. 88, 016802
(2002).
[4] B. Tanatar and D. M. Ceperley Phys. Rev. B 39, 5005 (1989).
[5] D. Vollhardt, Rev. Mod. Phys. 56, 99 (1984).
[6] Z. Lenac and M. Sunjic, Phys. Rev. B 52, 11238 (1995).
[7] D. Tanasković, et al., Phys. Rev. Lett. 91, 066603 (2003).
[8] M. C. O. Aguiar et al., Europhys. Lett. 67, 226 (2004).
[9] J. Jaroszynski et al, Phys. Rev. Lett. 92, 226403 (2004).
[10] S. Pankov and V. Dobrosavljević, Phys. Rev. Lett. 94, 046402
(2005).
	References

  We consider an interaction-driven scenario for the two-dimensional
metal-insulator transition in zero magnetic field (2D-MIT), based on melting
the Wigner crystal through vacancy-interstitial pair formation. We show that
the transition from the Wigner-Mott insulator to a heavy Fermi liquid emerges
as an instability to self-doping, resembling conceptually the solid to normal
liquid transition in He3. The resulting physical picture naturally explains
many puzzling features of the 2D-MIT.

2D-MIT as self-doping of aWigner-Mott insulator
S. Pankov ∗ V. Dobrosavljevic
National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306
Abstract
We consider an interaction-driven scenario for the two-dimensional metal-insulator transition in zero magnetic field (2D-MIT),
based on melting the Wigner crystal through vacancy-interstitial pair formation. We show that the transition from the Wigner-
Mott insulator to a heavy Fermi liquid emerges as an instability to self-doping, resembling conceptually the solid to normal liquid
transition in He3. The resulting physical picture naturally explains many puzzling features of the 2D-MIT.
Key words: Strong correlation; disorder; metal-insulator transition; Hubbard model
PACS: 71.27.+a, 72.15.Rn, 71.30.+h
A series of fascinating experiments, as first performed
some ten years ago by Kravchenko and co-workers [1], have
deeply changed our thinking about the two dimensional
electron gas (2DEG). These and many later experiments
demonstrated convincingly that the 2DEG can exhibit typ-
ical metallic behavior [2] above a well defined critical den-
sity nc. One of the most prominent features of this metallic
phase is an unprecedented resistivity drop, which is found
only in the low density regime n & nc close to the tran-
sition. These findings suggested that a well-defined metal-
insulator transition (MIT) may exist even in two dimen-
sions, in contrast to long held-beliefs based on the theories
for noninteracting disordered electrons.
These experiments are typically performed at such low
electron density where the relative strength of the Coulomb
interaction is so large (rs & 10), that the localization pro-
cesses could conceivably be suppressed by interaction ef-
fects. A diffusion-mode theory describing such interaction
renormalizations at weak disorder has been developed by
Finkelshtein and Punnoose [3], suggesting that sufficiently
strong interactions may indeed stabilize the metallic phase.
However, this theory can provide guidance only within a
narrowdiffusive regime restricted to very low temperatures.
In contrast, the most striking experimental results have
been established in a broad parameter range well outside
this regime, and are most pronounced in the cleanest sam-
ples [2]. In particular, the best established experimental sig-
nature of the transition relies on careful effective mass mea-
surements, which is found to diverges as m∗ ∝ (n− nc)
∗ Corresponding author; email: pankov@magnet.fsu.edu
1 1.05 1.1
-1.002
-0.998
-0.996
Fig. 1. Evolution of the free energy profile W [δ], as the electron
density is increased across the MIT (from top to bottom). The cusp
corresponds to the Wigner solid. The self-doped transition (blue
line) takes place before the instability emerges at half-filling (δ = 0,
orange line) and the Wigner insulating state is replaced by a heavy
Fermi liquid.
while the Lande g∗-factor remains largely unrenormalized.
Such phenomena cannot be understood within a low-energy
diffusion mode theory, which relies on Anderson localiza-
tion to produce an insulating phase.
A fundamental question is thus posed by these experi-
ments: what is the basic mechanism that drives the metal-
insulator transition in these systems? Does one have to rely
on disorder effects at all in the zero-th order approximation,
or can one understand the most important experimental
features by interaction effects alone? In this work, we con-
centrate solely on the effects of strong Coulomb interactions
in the clean limit, and try to establish which experimental
Preprint submitted to Elsevier 31 October 2018
http://arxiv.org/abs/0704.0250v1
features can be explained by entirely ignoring the disorder.
We approach the transition from the insulating side,
starting with the Wigner-Mott insulator, and examine
its melting by quantum fluctuations as density increases.
These are believed to be dominated [4] by the vacancy-
interstitial pair excitations on top of the classical triangu-
lar lattice configuration. Within the Wigner crystal, the
electrons are tightly bound, and the vacancy-interstitial
excitations can be well represented by a mapping to a two-
band Hubbard (i.e. charge-transfer) model, respectively
corresponding to the lattice and the interstitial electrons.
As the density increases, the charge-transfer gap eventu-
ally closes, and a metal-insulator transition assumes the
character of a Mott-metal-insulator transition, leading to
a strongly correlated metallic state on the metallic side.
Our model Hamiltonian reads:
fiσ + ecc
ciσ −
iσciσ + c
iσfiσ) +
fi↓ (1)
where f †, f and c†, c are creation and annihilation opera-
tors for site and interstitial electrons respectively. It is as-
sumed that the inter-cell hopping tij exists only between
interstitial orbitals, while only the site electrons are subject
to onsite Coulomb repulsion U , and the interstitial orbitals
are coupled to the site orbitals via hybridization V . The
local electrostatic potentials for the two bands are denoted
by ef and ec.
Wemodel the strong onsite repulsion by exclusion of dou-
ble ocupancy, using the standard slave-boson mean-field
formalism. The free energy per electron then reads:
W [λ, Z, µ, δ] =
ln (1 + exp (−(ε̃lk − µ)/T ))
(Z − 1) + µ (2)
where εlk are renormalized band energies, λ is the Lagrange
multiplier enforcing the slave-boson constraint, Z is the
quasiparticle weight, µ is the chemical potential. The chem-
ical potential µ is an internal parameter here, arising simi-
larly to λ, from constraining the electron density per unit
area. The self-doping δ measures deviation in the number
of electrons (per elementary cell) from the half filling. In
the classical limit of low electron density δ = 0, but it be-
comes δ 6= 0 at higher density, where quantum effects are
important. This is reminiscent of the liquid solid transition
in He3 [5]. The equations of the state are given by the sad-
dle point of the free energy W [λ, Z, µ, δ].
A peculiarity of this model is that the band energies
are not fixed, but are self-consistently determined through
their dependence on the occupation of the site and inter-
stitial orbitals. It immediately follows that the system is
unstable to the self-doping at the MIT. Indeed, by care-
fully accounting for the electrostatic energy balance due to
such charge transfer, we find that the free energy takes a
lower value (relative to the classical value at δ = 0) on one
of the branches at δ 6= 0 before the transition at half filling
is found (see Fig. 1).
With appropriately chosen model parameters we can ex-
plain the basic experimental results, which otherwise can-
not be captured by the diffusion mode theory. Similarly
to what is seen experimentally, we observe strong renor-
malization of the effective mass near the transition m∗ ∼
(n − nc)
−1. In this strongly correlated regime any extrin-
sic disorder is very effectively screened by interaction ef-
fects [7], providing a plausible scenario for the large resis-
tivity drop [8]. The magneto-resistance data are also natu-
rally explained by our two band model. The experiment [9]
has shown that in high parallel magnetic fields the trans-
port takes place by activated processes, with an activation
gap that vanishes linearly at some density n1 > nc. In the
strong field our model reduces to a trivial model of non-
interacting spinless electrons. For the density n < n1 the
system is a band insulator with the lower band completely
filled. The bands broaden as the density increases, and the
insulating gap closes linearly at some density n1 > nc.
Our static lattice model does not capture the collective
charge density fluctuations – the phonons of the Wigner
crystal. Because the Coulomb interaction is long ranged,
these collective modes are very soft, and play an important
role in renormalizing the model parameters. It is this strong
renormalization of the charge transfer gap ec − ef that
pushes the transition to such low density [6] (rs ≫ 1)), in a
fashion that is conceptually very similar to the formation of
the Coulomb gap [10]. At present, these effects are incorpo-
rated in through the choice of the electrostatic parameters
of our model. In future work, we would like to systemati-
cally incorporate these soft collective modes, the effects of
which are currently included in a semi-phenomenological
fashion. This program can be achieved by a variety of meth-
ods, including extended dynamical mean field approaches
(EDMFT), by exploiting unique properties of the Coulomb
potential, and by employing the techniques recently devel-
oped in the Coulomb glass context [10].
References
[1] S. V. Kravchenko et al., Phys. Rev. B 51, 7038 (1995).
[2] For a review, see S. V. Kravchenko and M. P. Sarachik, Rep.
Prog. Phys. 67, 1 (2004).
[3] A. Punnoose and A. M. Finkelstein, Phys. Rev. Lett. 88, 016802
(2002).
[4] B. Tanatar and D. M. Ceperley Phys. Rev. B 39, 5005 (1989).
[5] D. Vollhardt, Rev. Mod. Phys. 56, 99 (1984).
[6] Z. Lenac and M. Sunjic, Phys. Rev. B 52, 11238 (1995).
[7] D. Tanasković, et al., Phys. Rev. Lett. 91, 066603 (2003).
[8] M. C. O. Aguiar et al., Europhys. Lett. 67, 226 (2004).
[9] J. Jaroszynski et al, Phys. Rev. Lett. 92, 226403 (2004).
[10] S. Pankov and V. Dobrosavljević, Phys. Rev. Lett. 94, 046402
(2005).
	References
