1. Interplay of disorder and interactions
in two-dimensional semiconductors
Sergey Kravchenko
in collaboration with:
S. Anissimova V. T. Dolgopolov A. M. Finkelstein T. M. Klapwijk
NEU ISSP Weizmann TU Delft
A. Punnoose M. P. Sarachik A. A. Shashkin
CCNY CCNY ISSP
11/10/2018
SCCS 2008

2. Outline
Scaling theory of localization: “all electrons are localized in 2D”
Samples
What do experiments show?
Magnetic properties of strongly coupled electrons in 2D: ballistic regime (no disorder)
Interplay between disorder and interactions in 2D; flow diagram
Conclusions
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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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4. strength of interactions increases
Coulomb energy
Fermi energy
rs =
Gas Strongly correlated liquid Wigner crystal
Insulator ??????? Insulator
strength of interactions increases
~ ~ rs
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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)
Local moments, strong insulator
disorder
electron density
Local moments, 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
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6. What do experiments show?
Scaling theory of localization: “all electrons are localized in two dimensions
Samples
What do experiments show?
Magnetic properties of strongly coupled electrons in 2D: ballistic regime (no disorder)
Interplay between disorder and interactions in 2D; flow diagram
Conclusions
11/10/2018

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)
11/10/2018
SCCS 2008

8. Why Si MOSFETs? large m*= 0.19 m0 two valleys
low average dielectric constant e=7.7
As a result, at low electron densities, Coulomb energy strongly exceeds Fermi energy: EC >> EF
rs = EC / EF >10 can easily be reached in clean samples
11/10/2018
SCCS 2008

9. What do experiments show?
Scaling theory of localization: “all electrons are localized in two dimensions
Samples
What do experiments show?
Magnetic properties of strongly coupled electrons in 2D: ballistic regime (no disorder)
Interplay between disorder and interactions in 2D; flow diagram
Conclusions
11/10/2018

10. Strongly disordered Si MOSFET
(Pudalov et al.)
Consistent (more or less) with the one-parameter scaling theory
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11. Clean sample, much lower electron densities
S.V. Kravchenko, G.V. Kravchenko, W. Mason, J. Furneaux, V.M. Pudalov, and M. D’Iorio, PRB 1995
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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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13. The effect of the parallel magnetic field:
T = 30 mK
Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRL 2001
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14. 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)
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15. Scaling theory of localization: “all electrons are localized in 2D”
Samples
What do experiments show?
Magnetic properties of strongly coupled electrons in 2D: ballistic regime (no disorder)
Interplay between disorder and interactions in 2D; flow diagram
Conclusions
11/10/2018

16. Magnetic field of full spin polarization vs. electron density:
data become T-dependent
electron density (1011 cm-2)
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17. insulator T-dependent regime
Spin susceptibility exhibits critical behavior near the
sample-independent critical density nc : c ~ ns/(ns – nc)
insulator
T-dependent
regime
Are we approaching a phase transition?
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18. g-factor or effective mass?
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19. Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRB 66, 073303 (2002)
Effective mass vs. g-factor
Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRB 66, (2002)
Not Stoner scenario! Wigner crystal?
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20. 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, PRB (2007)
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21. disorder electron density Anderson insulator
Disorder increases at low density and we enter “Punnoose-Finkelstein regime”
disorder
paramagnetic Fermi-liquid
Wigner crystal?
Liquid ferromagnet?
Density-independent disorder
electron density
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22. Scaling theory of localization: “all electrons are localized in 2D”
Samples
What do experiments show?
Magnetic properties of strongly coupled electrons in 2D: ballistic regime (no disorder)
Interplay between disorder and interactions in 2D; flow diagram
Conclusions
11/10/2018

23. 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
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24. 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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25. 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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26. 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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27. Experimental results (low-disordered Si MOSFETs;
“just metallic” regime; ns= 9.14x1010 cm-2):
S. Anissimova et al., Nature Phys. 3, 707 (2007)
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28. 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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29. 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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30. 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)
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31. 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)

32. 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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33. Comparison between theory (lines) and experiment (symbols)
(no adjustable parameters used!)
S. Anissimova et al., Nature Phys. 3, 707 (2007)
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34. g-factor grows as T decreases
ns = 9.9 x 1010 cm-2
“ballistic” value
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35. SUMMARY:
Strong interactions in clean two-dimensional systems lead to strong increase and possible divergence of the spin susceptibility: the behavior characteristic of a phase transition
Disorder-interactions flow diagram of the metal-insulator transition clearly reveals a quantum critical point: i.e., there exists a metallic state and a metal-insulator transition in 2D, contrary to the 20-years old paradigm!
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