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

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 11/10/2018

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) 11/10/2018

strength of interactions increases Coulomb energy Fermi energy rs = Gas Strongly correlated liquid Wigner crystal Insulator ??????? Insulator strength of interactions increases ~1 ~35 rs 11/10/2018

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, 256601 (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 11/10/2018

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

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

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

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

Strongly disordered Si MOSFET (Pudalov et al.) Consistent (more or less) with the one-parameter scaling theory 11/10/2018

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 11/10/2018

In very clean samples, the transition is practically universal: Klapwijk’s sample: Pudalov’s sample: (Note: samples from different sources, measured in different labs) 11/10/2018

The effect of the parallel magnetic field: T = 30 mK Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRL 2001 11/10/2018

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) 11/10/2018

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

Magnetic field of full spin polarization vs. electron density: data become T-dependent electron density (1011 cm-2) 11/10/2018

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? 11/10/2018

g-factor or effective mass? 11/10/2018

Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRB 66, 073303 (2002) Effective mass vs. g-factor Shashkin, Kravchenko, Dolgopolov, and Klapwijk, PRB 66, 073303 (2002) Not Stoner scenario! Wigner crystal? 11/10/2018

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) 11/10/2018

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 11/10/2018

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

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 11/10/2018

Effective strength of interactions grows as the temperature decreases Zeitschrift fur Physik B (Condensed Matter) -- 1984 -- vol.56, no.3, pp. 189-96 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 11/10/2018

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 11/10/2018

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, 11/10/2018

Experimental results (low-disordered Si MOSFETs; “just metallic” regime; ns= 9.14x1010 cm-2): S. Anissimova et al., Nature Phys. 3, 707 (2007) 11/10/2018

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 11/10/2018

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) 11/10/2018

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) 11/10/2018

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 11/10/2018 rmax ln(T/Tmax)

Resistance and interactions vs. T Note that the metallic behavior sets in when g2 ~ 0.45, exactly as predicted by the RG theory 11/10/2018

Comparison between theory (lines) and experiment (symbols) (no adjustable parameters used!) S. Anissimova et al., Nature Phys. 3, 707 (2007) 11/10/2018

g-factor grows as T decreases ns = 9.9 x 1010 cm-2 “ballistic” value 11/10/2018

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! 11/10/2018