1. Effects of electron-electron interactions
in two dimensions
Sergey Kravchenko
in collaboration with: S. Anissimova, V.T. Dolgopolov, A. M. Finkelstein, T.M. Klapwijk, A. Punnoose, A.A. Shashkin
11/24/2018
Hsinchu 2007

2. Outline
Do interactions modify “all states are localized in 2D” paradigm?
(or: what happens to the Anderson localization in the presence of interactions?)
Samples
What do experiments show?
“Clean” regime: diverging spin susceptibility
“Dirty” regime: interplay between disorder and interactions
Summary
11/24/2018
Hsinchu 2007

3. However, later this prediction was shown to be incorrect
Corrections to conductivity due to electron-electron interactions
in the diffusive regime (Tt < 1)
always insulating behavior
However, later this prediction was shown to be incorrect
11/24/2018
Hsinchu 2007

4. Effective strength of interactions grows as the temperature decreases
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 & Castellani-DiCastro-Lee-Ma’s term
11/24/2018
Hsinchu 2007

5. Same mechanism persists to ballistic regime (Tt > 1),
but corrections become linear in temperature
This is reminiscent of earlier Stern-Das Sarma’s result
where C(ns) < 0
(However, Das Sarma’s calculations are not applicable to strongly interacting regime because at r s>1, the screening length becomes smaller than the separation between electrons.)
11/24/2018
Hsinchu 2007

6. Do interactions modify “all states are localized in 2D” paradigm?
(or: what happens to the Anderson localization in the presence of interactions?)
Samples
What do experiments show?
“Clean” regime: diverging spin susceptibility
“Dirty” regime: interplay between disorder and interactions
Summary
11/24/2018
Hsinchu 2007

7. distance into the sample (perpendicular to the surface)
silicon MOSFET Al
SiO2
p-Si
conductance band
2D electrons
chemical potential
energy
valence band _ +
distance into the sample (perpendicular to the surface)
11/24/2018
Hsinchu 2007

8. Why Si MOSFETs? large m*= 0.19 m0 two valleys
low average dielectric constant e=7.7
As a result, at low densities, Coulomb energy strongly exceeds Fermi energy: EC >> EF
rs = EC / EF >10 can easily be reached in clean samples
11/24/2018
Hsinchu 2007

9. Do interactions modify “all states are localized in 2D” paradigm?
(or: what happens to the Anderson localization in the presence of interactions?)
Samples
What do experiments show?
“Clean” regime: diverging spin susceptibility
“Dirty” regime: interplay between disorder and interactions
Summary
11/24/2018
Hsinchu 2007

10. Strongly disordered Si MOSFET
(Pudalov et al.)
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Hsinchu 2007

11. Clean sample, much lower electron densities
Kravchenko, Mason, Bowker, Furneaux, Pudalov, and D’Iorio, PRB 1995
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Hsinchu 2007

12. Similar transition is also observed in other 2D structures:
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, Tsui’s group)
p-AlAs (Shayegan’s group)
Hanein, Shahar, Tsui et al., PRL 1998
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Hsinchu 2007

13. In very clean samples, the transition is practically universal:
Klapwijk’s sample:
Pudalov’s sample:
(Note: samples from different sources, measured in different labs)
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Hsinchu 2007

14. … in contrast to strongly disordered samples:
clean sample:
disordered sample:
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Hsinchu 2007

15. The effect of magnetic field
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Hsinchu 2007

16. T = 30 mK Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRL 2001
11/24/2018
Hsinchu 2007

17. Magnetic field, by aligning spins, changes metallic R(T) to insulating:
Such a dramatic reaction on parallel magnetic field suggests unusual magnetic properties
(spins aligned)
11/24/2018
Hsinchu 2007

18. Do interactions modify “all states are localized in 2D” paradigm?
(or: what happens to the Anderson localization in the presence of interactions?)
Samples
What do experiments show?
“Clean” regime: diverging spin susceptibility
“Dirty” regime: interplay between disorder and interactions
Summary
11/24/2018
Hsinchu 2007

19. How to study magnetic properties of 2D electrons?
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Hsinchu 2007

20. Method 1: 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)
11/24/2018
Hsinchu 2007

21. Spontaneous spin polarization at nc?
Extrapolated polarization field, Bc,
vanishes at a finite electron density, nc
Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRL 2001 nc
Spontaneous spin polarization at nc?
11/24/2018
Hsinchu 2007

22. Comparison to other groups’ data
Shashkin et al, 2001
Pudalov et al, 2002
Vitkalov, Sarachik et al, 2001 nc
cm-2 is sample-independent
11/24/2018
Hsinchu 2007

23. Method 2: measurements of thermodynamic magnetization
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 + dB
2D electron gas
Ohmic contact
i ~ dm/dB = - dM/dns
11/24/2018
Hsinchu 2007

24. Magnetization of non-interacting electrons
spin-down
spin-up
gmBB dM M
dns mB ns ns
11/24/2018
Hsinchu 2007

25. Magnetic field of the full spin polarization vs. ns
spontaneous spin polarization at nc:
non-interacting system
mBns B/Bc for B < Bc
Bc = ph2ns/mB g*m*
Bc = ph2ns/2mBmb
M = mBx ns =
mBns for B > Bc dM Bc
dns
B > Bc B ns nc
B < Bc ns
11/24/2018
Hsinchu 2007

26. Raw magnetization data: induced current vs. gate voltage
dm/dB = - dM/dn
Raw magnetization data: induced current vs. gate voltage
1 fA!!
B|| = 5 tesla
11/24/2018
Hsinchu 2007

27. 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
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Hsinchu 2007

28. Summary of the results obtained by four independent methods (including transport)
11/24/2018
Hsinchu 2007

29. insulator T-dependent regime
Spin susceptibility exhibits critical behavior near the
metal-insulator transition: c ~ ns/(ns – nc)
insulator
T-dependent
regime
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Hsinchu 2007

30. Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRB 66, 073303 (2002)
Effective mass vs. g-factor
Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRB 66, (2002)
11/24/2018
Hsinchu 2007

31. Effective mass as a function of rs-2 in Si(111) and Si(100)
Si(111): peak mobility 2.5x103 cm2/Vs
Si(100): peak mobility 3x104 cm2/Vs
Si (100)
Shashkin, Kapustin, Deviatov, Dolgopolov, and Kvon, in preparation
11/24/2018
Hsinchu 2007

32. disorder electron density Anderson insulator
Disorder increases at low density due to reduced screening
disorder
paramagnetic Fermi-liquid
Wigner crystal?
Liquid ferromagnet?
Density-independent disorder
electron density
11/24/2018
Hsinchu 2007

33. Do interactions modify “all states are localized in 2D” paradigm?
(or: what happens to the Anderson localization in the presence of interactions?)
Samples
What do experiments show?
“Clean” regime: diverging spin susceptibility
“Dirty” regime: interplay between disorder and interactions
Summary
11/24/2018
Hsinchu 2007

34. Recent development: two-loop RG theory
11/24/2018
Hsinchu 2007

35. metallic phase stabilized
disorder takes over
disorder
QCP
interactions
Punnoose and Finkelstein, Science
310, 289 (2005)
metallic phase stabilized
by e-e interaction
11/24/2018
Hsinchu 2007

36. 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,
11/24/2018
Hsinchu 2007

37. Experimental results (low-disordered Si MOSFETs;
“just metallic” regime; ns= 9.14x1010 cm-2):
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Hsinchu 2007

38. Temperature dependences of the
resistance (a) and strength of interactions (b)
This is the first time effective strength of interactions
has been seen to depend on T
11/24/2018
Hsinchu 2007

39. Experimental disorder-interaction flow diagram of the 2D electron liquid
11/24/2018
Hsinchu 2007

40. Experimental vs. theoretical flow diagram (qualitative comparison b/c the 2-loop theory was developed for multi-valley systems)
11/24/2018
Hsinchu 2007

41. 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
11/24/2018
Hsinchu 2007
rmax ln(T/Tmax)

42. Resistance and interactions vs. T
Note that the metallic behavior sets in when g2 ~ 0.45,
exactly as predicted by the RG theory
11/24/2018
Hsinchu 2007

43. Comparison between theory (lines) and experiment (symbols)
(no adjustable parameters used!)
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Hsinchu 2007

44. g-factor grows as T decreases
ns = 9.9 x 1010 cm-2
“ballistic” value
11/24/2018
Hsinchu 2007

45. SUMMARY:
In the clean (ballistic) regime, Pauli spin susceptibility critically grows with a tendency to diverge near a certain electron density nc suggesting the existence of a magnetic phase transition.
However, upon approaching to nc, one leaves the clean regime and enters the “Punnoose-Finkelstein” regime where the physics is governed by interplay between interactions and disorder. In this regime, both interactions and disorder become temperature-dependent.
Punnoose-Finkelstein theory gives quantitatively correct description of the metal-insulator transition. In particular, in excellent agreement with theory, the metallic behavior sets in once g2 > 0.45!
11/24/2018
Hsinchu 2007