1. Interactions and disorder in two dimensions
Sergey Kravchenko
in collaboration with:
S. Anissimova, V.T. Dolgopolov, A. M. Finkelstein, T.M. Klapwijk,
A. Punnoose, A.A. Shashkin
1/16/2019
Leiden 2007

2. Outline
Scaling theory of localization: the origin of the common wisdom “all electron states are localized in 2D”
Samples
What do experiments show?
What do theorists have to say?
Interplay between disorder and interactions: experimental test
“Clean” regime: diverging spin susceptibility
Summary
1/16/2019
Leiden 2007

3. One-parameter scaling theory for non-interacting electrons:
the origin of the common wisdom “all states are localized in 2D”
d(lnG)/d(lnL) = b(G)
G ~ Ld-2 exp(-L/Lloc)
QM interference
Ohm’s law in d dimensions
metal (dG/dL>0)
insulator
G = 1/R
insulator L
insulator (dG/dL<0)
Abrahams, Anderson, Licciardello, and Ramakrishnan, PRL 42, 673 (1979)
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Leiden 2007

4. Solution (Pruisken, Khmelnitskii…): Two-parameter scaling theory
However, the existence of the quantum Hall effect (discovered in one year later) is inconsistent with this prediction
... I remember being challenged over that well-known fact that all states were localized in two dimensions, something that made no sense at all in light of the experiments I had just shown.
R. B. Laughlin, Nobel Lecture
Solution (Pruisken, Khmelnitskii…):
Two-parameter scaling theory
1/16/2019
Leiden 2007

5. What do experiments show? What do theorists have to say?
Scaling theory of localization: the origin of the common wisdom “all electron states are localized in 2D”
Samples
What do experiments show?
What do theorists have to say?
Interplay between disorder and interactions: experimental test
“Clean” regime: diverging spin susceptibility
Summary
1/16/2019
Leiden 2007

6. 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)
1/16/2019
Leiden 2007

7. Why Si MOSFETs?
It turns out to be a very convenient 2D system to study strongly-interacting regime because of:
large effective mass m*= 0.19 m0
two valleys in the electronic spectrum
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
1/16/2019
Leiden 2007

8. What do experiments show? What do theorists have to say?
Scaling theory of localization: the origin of the common wisdom “all electron states are localized in 2D”
Samples
What do experiments show?
What do theorists have to say?
Interplay between disorder and interactions: experimental test
“Clean” regime: diverging spin susceptibility
Summary
1/16/2019
Leiden 2007

9. Strongly disordered Si MOSFET
(Pudalov et al.)
Consistent (more or less) with the one-parameter scaling theory
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Leiden 2007

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

11. 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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Leiden 2007

12. 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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Leiden 2007

13. … in contrast to strongly disordered samples:
clean sample:
disordered sample:
Clearly, one-parameter scaling theory does not work here
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Leiden 2007

14. The effect of the parallel magnetic field:
T = 30 mK
Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRL 2001
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Leiden 2007

15. Magnetic field, by aligning spins, changes metallic R(T) to insulating:
(spins aligned)
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Leiden 2007

16. What do experiments show? What do theorists have to say?
Scaling theory of localization: the origin of the common wisdom “all electron states are localized in 2D”
Samples
What do experiments show?
What do theorists have to say?
Interplay between disorder and interactions: experimental test
“Clean” regime: diverging spin susceptibility
Summary
1/16/2019
Leiden 2007

17. However, later this prediction was shown to be incorrect
Corrections to conductivity due to electron-electron interactions
in the diffusive regime (Tt < 1)
(0<F<1, N.B.: F is not the Landau Fermi-liquid constant)
always insulating behavior
However, later this prediction was shown to be incorrect
1/16/2019
Leiden 2007

18. 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
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Leiden 2007

19. 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 rs>1, the screening length becomes smaller than the separation between electrons.)
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Leiden 2007

20. Recent theory of the MIT in 2D
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Leiden 2007

21. metallic phase stabilized
disorder takes over
disorder
QCP
interactions
Punnoose and Finkelstein, Science
310, 289 (2005)
metallic phase stabilized
by e-e interaction
1/16/2019
Leiden 2007

22. What do experiments show? What do theorists have to say?
Scaling theory of localization: the origin of the common wisdom “all electron states are localized in 2D”
Samples
What do experiments show?
What do theorists have to say?
Interplay between disorder and interactions: experimental test
“Clean” regime: diverging spin susceptibility
Summary
1/16/2019
Leiden 2007

23. - diffusive: low temperatures, higher disorder (Tt < 1).
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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Leiden 2007

24. Experimental results (low-disordered Si MOSFETs;
“just metallic” regime; ns= 9.14x1010 cm-2):
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25. 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
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26. Experimental disorder-interaction flow diagram of the 2D electron liquid
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27. Experimental vs. theoretical flow diagram (qualitative comparison b/c the 2-loop theory was developed for multi-valley systems)
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Leiden 2007

28. 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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Leiden 2007
rmax ln(T/Tmax)

29. 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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Leiden 2007

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

31. g-factor grows as T decreases
ns = 9.9 x 1010 cm-2
“non-interacting” value
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Leiden 2007

32. What do experiments show? What do theorists have to say?
Scaling theory of localization: the origin of the common wisdom “all electron states are localized in 2D”
Samples
What do experiments show?
What do theorists have to say?
Interplay between disorder and interactions: experimental test
“Clean” regime: diverging spin susceptibility
Summary
1/16/2019
Leiden 2007

33. How to study magnetic properties of 2D electrons?
1/16/2019
Leiden 2007

34. 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)
1/16/2019
Leiden 2007

35. 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?
1/16/2019
Leiden 2007

36. Method 2: weak-field Shubnikov-de Haas oscillations
high density
low density
(Pudalov et al., PRL 2002; Shashkin et al, PRL 2003)
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Leiden 2007

37. Comparison to other groups’ data
Shashkin et al, 2001
Pudalov et al, 2002
Vitkalov, Sarachik et al, 2001 nc
cm-2 is sample-independent
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Leiden 2007

38. Method 3: 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
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Leiden 2007

39. Magnetization of non-interacting electrons
spin-down
spin-up
gmBB dM M
dns mB ns ns
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Leiden 2007

40. 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
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Leiden 2007

41. 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
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Leiden 2007

42. 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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Leiden 2007

43. Summary of the results obtained by four independent methods (including transport)
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Leiden 2007

44. insulator T-dependent regime
Spin susceptibility exhibits critical behavior near the
sample-independent critical density nc : c ~ ns/(ns – nc)
insulator
T-dependent
regime
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Leiden 2007

45. disorder electron density Anderson insulator
Disorder increases at low density and we enter “Finkelstein regime”
disorder
paramagnetic Fermi-liquid
Wigner crystal?
Liquid ferromagnet?
Density-independent disorder
electron density
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Leiden 2007

46. Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRB 66, 073303 (2002)
Effective mass vs. g-factor
Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRB 66, (2002)
1/16/2019
Leiden 2007

47. 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, preprint
1/16/2019
Leiden 2007

48. SUMMARY:
In the clean (ballistic) regime, Pauli spin susceptibility critically grows with a tendency to diverge near a certain electron density nc suggesting a quantum phase transition at ns=nc
However, upon approaching to nc, one leaves the clean regime and enters the diffusive regime, where the physics is governed by the interplay between interactions and disorder. In this regime, both interactions and disorder become temperature-dependent
RG theory gives quantitatively correct description of the metallic state. In particular, in excellent agreement with theory, the metallic behavior sets in once g2 > 0.45!
1/16/2019
Leiden 2007