1. Metal-Insulator Transition and Related Phenomena in 2D
Sergey Kravchenko
in collaboration with:
S. Anissimova V. T. Dolgopolov A. M. Finkelstein T. M. Klapwijk
Formerly at NEU ISSP Texas A&M TU Delft
V. M. Pudalov A. Punnoose M. P. Sarachik A. A. Shashkin
Lebedev Institute CCNY CCNY ISSP 1 1

2. Outline Possible disorder-interactions phase diagrams in 2D Samples
What do transport experiments show?
Interplay between disorder and interactions in 2D; flow diagram
Spin susceptibility
g-factor or effective mass?
Is it a Wigner crystal?
Summary

3. In 1979, a powerful theory was created by the “Gang of Four”
(Abrahams, Anderson, Licciardello, and Ramakrishnan), according to which, there is no conductivity in 2D at zero temperature and in zero magnetic field.
This became one of the most influential paradigms in modern
condensed matter physics
(works for non-interacting electrons)

4. strength of interactions increases
Coulomb energy
Fermi energy
rs =
Wigner crystal Strongly correlated liquid Gas
strength of interactions increases
~ ~ rs
Terra incognita
Long range order
Short range order
Random electrons 4

5. Suggested phase diagrams for strongly interacting electrons in two dimensions
Tanatar and Ceperley, Phys. Rev. B 39, 5005 (1989)
Attaccalite et al. Phys. Rev. Lett. 88, (2002)
strong insulator
disorder
electron density
strong insulator
disorder
electron density
strongly disordered sample
Wigner
crystal
Ferromagnetic Fermi liquid
Paramagnetic Fermi liquid, weak insulator
Wigner
crystal
Paramagnetic Fermi liquid, weak insulator
clean sample
strength of interactions increases
strength of interactions increases

6. EF = (h2/m) Ns EC = (e2/ε) Ns1/2
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:
University of Virginia 6 6

7. Possible disorder-interactions phase diagrams in 2D
Samples
What do transport experiments show?
Interplay between disorder and interactions in 2D; flow diagram
Spin susceptibility
g-factor or effective mass?
Is it a Wigner crystal?
Summary

8. distance into the sample (perpendicular to the surface)Al
SiO2
p-Si
conductance band
2D electrons
chemical potential
energy
valence band _ +
distance into the sample (perpendicular to the surface)
10/10/09
University of Virginia 8 8

9. 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. 9

10. Possible disorder-interactions phase diagrams in 2D
Samples
What do transport experiments show?
Interplay between disorder and interactions in 2D; flow diagram
Spin susceptibility
g-factor or effective mass?
Is it a Wigner crystal?
Summary

11. This is what it is expected to look like (weakly-interacting electrons)…
(Pudalov et al.)

12. …but this is what it looks like when the
electron-electron interactions are strong
S.V.K., Mason, Bowker, Furneaux, Pudalov, and D’Iorio, PRB 1995 12 12

13. In very clean samples, the transition is practically universal:
S.V.K. and Klapwijk, PRL 2000
Sarachik and S.V.K., PNAS 1999
Klapwijk’s sample
Pudalov’s sample 13

14. Reaction of referees (1993):
Referee A:
“The paper should not be published in PRL. Everyone knows there is no zero-temperature conductivity in 2-d.”
Referee B:
“The reported results are most intriguing, but they must be wrong.
If there indeed were a metal-insulator transition in these systems,
it would have been discovered years ago.”
Referee C:
“I cannot explain the reported behavior offhand. Therefore, it must
be an experimental error.”

15. Timeline: However, also in 1997….
1993: Metal-insulator transition in 2D is discovered.
Paper submitted to Phys. Rev. Lett. and rejected.
Proposal submitted to NSF and declined.
1994: Proposal submitted to NSF and declined.
1995: Proposal submitted to NSF and declined.
1996: Proposal submitted to NSF and declined.
1997: Proposal submitted to NSF and declined.
However, also in 1997….

16. …a similar transition has been confirmed in Si MOSFETs from another source and observed in other 2D structures:
Si-MOSFETS from IBM (Popovic, Washburn, and Fowler)
p-Si:Ge (Coleridge’s group; Ensslin’s group)
p-GaAs/AlGaAs (Tsui’s group, Boebinger’s group)
n-GaAs/AlGaAs (Tsui’s group, Stormer’s group, Eisenstein’s group)
n-Si:Ge (Okamoto’s group)
p-AlAs (Shayegan’s group)
(Hanein, Shahar, Tsui et al., PRL 1998) 16

17. Possible disorder-interactions phase diagrams in 2D
Samples
What do transport experiments show?
Interplay between disorder and interactions in 2D; flow diagram
Spin susceptibility
g-factor or effective mass?
Is it a Wigner crystal?
Summary

18. always insulating behavior
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19. Altshuler-Aronov-Lee’s result
Zeitschrift fur Physik B (Condensed Matter) vol.56, no.3, pp
Weak localization and Coulomb interaction in disordered systems
Finkel'stein, A.M.
L.D. Landau Inst. for Theoretical Phys., Acad. of Sci., Moscow, USSR
Insulating behavior when interactions are weak
Metallic behavior when interactions are strong
Effective strength of interactions grows as the temperature decreases
Altshuler-Aronov-Lee’s result
Finkelstein’s term

20. More recent development: two-loop RG theory metallic phase stabilized
disorder takes over
disorder
QCP
interactions
Punnoose and Finkelstein, Science
310, 289 (2005)
metallic phase stabilized
by e-e interaction
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21. Experimental test
First, one needs to ensure that the system is in the diffusive regime (Tt < 1). One can distinguish between diffusive and ballistic regimes by studying magnetoconductance:
- diffusive: low temperatures, higher disorder (Tt < 1).
- ballistic: low disorder, higher temperatures (Tt > 1).
The exact formula for magnetoconductance (Lee and Ramakrishnan, 1982):
2 valleys
for
Low-field magnetoconductance in the diffusive regime yields strength of electron-electron interactions
In standard Fermi-liquid notations,
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22. Temperature dependences of the
resistance (a) and strength of interactions (b)
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23. S. Anissimova et al., Nature Phys. 3, 707 (2007)
Experimental disorder-interaction flow diagram of the 2D electron liquid
S. Anissimova et al., Nature Phys. 3, 707 (2007)
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24. S. Anissimova et al., Nature Phys. 3, 707 (2007)
Experimental vs. theoretical flow diagram (qualitative comparison b/c the 2-loop theory was developed for multi-valley systems)
S. Anissimova et al., Nature Phys. 3, 707 (2007)
9/21/2018

25. Solutions of the RG-equations for r << ph/e2:
Quantitative predictions of the one-loop RG for 2-valley systems (Punnoose and Finkelstein, Phys. Rev. Lett. 2002)
Solutions of the RG-equations for r << ph/e2:
a series of non-monotonic curves r(T). After rescaling, the solutions are described by a single universal curve:
rmax
r(T)
Tmax
g2(T)
For a 2-valley system (like Si MOSFET),
metallic r(T) sets in when g2 > 0.45
g2 = 0.45
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rmax ln(T/Tmax)

26. Resistance and interactions vs. T
Note that the metallic behavior sets in when g2 ~ 0.45,
exactly as predicted by the RG theory
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27. Comparison between theory (lines) and experiment (symbols)
(no adjustable parameters used!)
S. Anissimova et al., Nature Phys. 3, 707 (2007)
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28. g-factor grows as T decreases
ns = 9.9 x 1010 cm-2
“ballistic” value
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29. Everything seems very nice! But…
r(T) curves in the ballistic regime look similar (except for there is no maximum), but cannot be explained by Finkelstein’s theory 29 29

30. Let’s look at the ballistic regime

31. Possible disorder-interactions phase diagrams in 2D
Samples
What do transport experiments show?
Interplay between disorder and interactions in 2D; flow diagram
Spin susceptibility
g-factor or effective mass?
Is it a Wigner crystal?
Summary

32. The effect of the parallel magnetic field:

33. Magnetic field, by aligning spins, changes metallic R(T) to insulating:
Such a dramatic reaction on parallel magnetic field suggests unusual spin properties
(spins aligned) 33

34. Magnetoresistance in a parallel magnetic field
T = 30 mK Bc Bc
Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRL 2001 Bc
Spins become fully polarized
(Okamoto et al., PRL 1999;
Vitkalov et al., PRL 2000)

35. Spontaneous spin polarization at n?
Extrapolated polarization field, Bc,
vanishes at a finite electron density, n
Shashkin, S.V.K., Dolgopolov, and Klapwijk, PRL 2001 n
Spontaneous spin polarization at n?

36. gm as a function of electron density calculated using g. m
gm as a function of electron density calculated using g*m* = ћ2ns / BcB
(Shashkin et al., PRL 2001) n

37. i ∝ d/dB = - dM/dns Magnetic measurements without magnetometer
suggested by B. I. Halperin (1998); first implemented by O. Prus, M. Reznikov, U. Sivan et al. (2002)
1010 Ohm - +
Gate Vg
Current amplifier
SiO2 Si
Modulated magnetic field
B + B
2D electron gas
Ohmic contact
i ∝ d/dB = - dM/dns 37

38. Raw magnetization data: induced current vs. gate voltage
d/dB = - dM/dn
1 fA!!
B|| = 5 tesla 38

39. Raw magnetization data: induced current vs. gate voltage
Integral of the previous slide gives M (ns):
complete spin polarization
at ns=1.5x1011 cm-2
B|| = 5 tesla
Bar-Ilan University 39

40. insulator T-dependent regime
Spin susceptibility exhibits critical behavior near the
sample-independent critical density n :  ∝ ns/(ns – n)
Critical behavior of a thermodynamic parameter suggests a phase transition!
insulator
T-dependent
regime 40

41. Possible disorder-interactions phase diagrams in 2D
Samples
What do transport experiments show?
Interplay between disorder and interactions in 2D; flow diagram
Spin susceptibility
g-factor or effective mass?
Is it a Wigner crystal?
Summary

42. Shashkin, S.V.K., Dolgopolov, and Klapwijk, PRB (2002)
Effective mass vs. g-factor
(from the analysis of the transport data in spirit of
Zala, Narozhny, and Aleiner, PRB 2001) :
Shashkin, S.V.K., Dolgopolov, and Klapwijk, PRB (2002)

43. (Rahimi, Anissimova, Sakr, S.V.K., and Klapwijk, PRL 2003)
Another way to measure m*: amplitude of the weak-field Shubnikov-de Haas oscillations vs. temperature
high density
low density
(Rahimi, Anissimova, Sakr, S.V.K., and Klapwijk, PRL 2003)

44. Comparison of the effective masses determined by two independent experimental methods: rs
(Shashkin, Rahimi, Anissimova, S.V.K., Dolgopolov, and Klapwijk, PRL 2003)

45. Yet another way to measure the effective mass: Thermopower
In the low-temperature metallic regime, the diffusion thermopower of strongly interacting 2D electrons is given by the relation
(Dolgopolov and Gold, 2011)

46. Divergence of thermopower
Thermopower tends to infinity with decreasing electron density at a finite density.

47. 1/S tends to vanish at nt
Thermopower tends to infinity with decreasing electron density at a finite density.

48. 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
As shown in Ref. [22], Eq. (1) holds for the strongly-interacting regime, the dependence of  on electron density is rather weak, and the main effect of electron-electron interactions is the suppression of the thermopower S.
The effective mass here is the effective electron mass at the Fermi level, and is different from the bandwidth related mass determined by parallel field magnetization measurements which is the Fermi energy counted from the band bottom [16] A. A. Shashkin et al., PRL 96, (2006); S. Anissimova et al., PRL 96, (2006)
(−T/S) is plotted as a function of ns: the data fall on a straight line with intercept (−T/S) → 0 at ns = nt .
The effective mass m∗ is obtained for the same samples by different measurements [A. A. Shashkin et al., PRB 66, (2002)].
Comparing the two data sets, they show similar behavior. However, the thermopower data do not yield the absolute value of m because of uncertainty in the coefficient a in Eq. (1)
(while the effective electron mass at the Fermi level tends to diverge, the bandwidth does not decrease appreciably in the close vicinity of the critical point nt)
Theoretical model: Extending the Fermi liquid concept to the strongly-interacting limit --- there is a clear distinction between the mass at the Fermi level and the bandwidth-related mass [V. A. Khodel et al, PRB 78, (2008)]
Mass enhancement m/mb in our experiment reaches around 25, very close to the critical point
Data of Fletcher, Pudalov et al.
Since S/T ∝ m/ns, divergence of the thermopower indicates a divergence of the effective mass:
m ∝ ns /(ns − nt)
We observe the increase of the effective mass up to m  25mb  5me!! 48

49. A divergence of the effective mass has been predicted…
using Gutzwiller's theory (Dolgopolov, JETP Lett. 2002)
solving an extended Hubbard model using dynamical mean-field theory (Pankov and Dobrosavljevic, PRB 2008)
from a renormalization group analysis for multi-valley 2D systems (Punnoose and Finkelstein, Science 2005)
by Monte-Carlo simulations (Marchi et al., PRB 2009; Fleury and Waintal, PRB 2010)
using an analogy with He3 near the onset of Wigner crystallization (Spivak and Kivelson, PRB 2004)
As shown in Ref. [22], Eq. (1) holds for the strongly-interacting regime, the dependence of  on electron density is rather weak, and the main effect of electron-electron interactions is the suppression of the thermopower S.
The effective mass here is the effective electron mass at the Fermi level, and is different from the bandwidth related mass determined by parallel field magnetization measurements which is the Fermi energy counted from the band bottom [16] A. A. Shashkin et al., PRL 96, (2006); S. Anissimova et al., PRL 96, (2006)
(−T/S) is plotted as a function of ns: the data fall on a straight line with intercept (−T/S) → 0 at ns = nt .
The effective mass m∗ is obtained for the same samples by different measurements [A. A. Shashkin et al., PRB 66, (2002)].
Comparing the two data sets, they show similar behavior. However, the thermopower data do not yield the absolute value of m because of uncertainty in the coefficient a in Eq. (1)
(while the effective electron mass at the Fermi level tends to diverge, the bandwidth does not decrease appreciably in the close vicinity of the critical point nt)
Theoretical model: Extending the Fermi liquid concept to the strongly-interacting limit --- there is a clear distinction between the mass at the Fermi level and the bandwidth-related mass [V. A. Khodel et al, PRB 78, (2008)]
Mass enhancement m/mb in our experiment reaches around 25, very close to the critical point 49

50. Transport properties of the insulator
V. M. Pudalov et al, PRL 1993

51. 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 experiment
Eth = 1 – 10 V/m !
De-pinning of a pinned Wigner
solid?
As shown in Ref. [22], Eq. (1) holds for the strongly-interacting regime, the dependence of  on electron density is rather weak, and the main effect of electron-electron interactions is the suppression of the thermopower S.
The effective mass here is the effective electron mass at the Fermi level, and is different from the bandwidth related mass determined by parallel field magnetization measurements which is the Fermi energy counted from the band bottom [16] A. A. Shashkin et al., PRL 96, (2006); S. Anissimova et al., PRL 96, (2006)
(−T/S) is plotted as a function of ns: the data fall on a straight line with intercept (−T/S) → 0 at ns = nt .
The effective mass m∗ is obtained for the same samples by different measurements [A. A. Shashkin et al., PRB 66, (2002)].
Comparing the two data sets, they show similar behavior. However, the thermopower data do not yield the absolute value of m because of uncertainty in the coefficient a in Eq. (1)
(while the effective electron mass at the Fermi level tends to diverge, the bandwidth does not decrease appreciably in the close vicinity of the critical point nt)
Theoretical model: Extending the Fermi liquid concept to the strongly-interacting limit --- there is a clear distinction between the mass at the Fermi level and the bandwidth-related mass [V. A. Khodel et al, PRB 78, (2008)]
Mass enhancement m/mb in our experiment reaches around 25, very close to the critical point 51

52. SUMMARY:
Competition between electron-electron interactions and disorder leads to the existence of the metal-insulator transition in two dimensions. The metallic state is stabilized by the electron-electron interactions. Disorder-interactions flow diagram of the metal-insulator transition reveals a quantum critical point.
In the clean (ballistic) 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. 52