1. Metal-Insulator Transition in 2D Electron Systems: Recent Progress Experiment: Dima Knyazev, Oleg Omel’yanovskii Vladimir Pudalov Theory: Igor Burmistrov, Nickolai Chtchelkatchev Schegolev memorial conference. Oct. 11-16, 2009 P.N. Lebedev Physical Institute, Moscow L.D. Landau Institute, Chernogolovka

2. Groundstate(s) of the 2D electron liquid (T  0) Major question to be addressed: Outline Historical intro: classical, semiclassical, quantum transport and 1-parameter scaling MIT in high mobility 2D systems The puzzle of the metallic-like conduction Quantifying e-e interaction in 2D Transport in the critical regime: 2 parameter RG theory Data analysis in the vicinity of the fixed point

3. 1.1. Classical charge transport 1. T >>h  D. Phonon scattering   1/T 2. T << h  D. Phonon scattering   1/T 5 3.T << T F. e-e scattering   1/T 2 4. T << T F. Impurity scattering   Const Note (a): Note (a): There is no σ(T) dependence in the T=0 limit ! (within the classical approximation, for non-interacting electrons ) + Umklapp

4. 1.2.Semiclassical concept of transport (1960) Ioffe-Regel criterion A.F. Ioffe and A.R. Regel, Prog. Semicond. 4, 237 (1960). Abram F. Ioffe Anatoly R. Regel “minimum metallic conductivity” Nevil Mott (1905-96)

5.  Possible behavior of resistivity (dimensionality is irrelevant): Semiclassical picture: MIT at T = 0 (1970’s)

6. All electrons in 2D become localized at T  0 1.3. Quantum concept of transport (1979): E.Abrahams T.V. Ramakrishnan A B Competition between dimensionality and interefrence Interference of electron waves causes localization for ln(1/T  )   Note (b) P.W. Anderson D.Khmelnitskii L.P.Gorkov

7. 1.4. Scaling ideas in the quantum transport picture: Thouless (1974, 77); Abrahams, Anderson, Licciardello, Ramakrishnan (’79); Wegner (’79). Renormalization Group transformation: The block size is increased from l tr to L 1-Parameter scaling equation At the MIT: g(L) – dimensionless conductance for a sample (size L) in units of e 2 /h For 2D system: β is always <0; there is no metallic state and no MIT

8. One-parameter scaling and experiment Note (c) Note (c) : The sign of dρ/dT at finite T is not indicative of the metallic or insulating state Low-mobility sample (μ=1.5  10 3 cm 2/ Vs) n

9. 2.Metal-insulator transition in high mobility 2D system S.Kravchenko, VP, et al., PRB 50, 8039 (1994) N ~10 11 cm -2 density  =4,5m 2 /Vs

10. Similar  (T) behavior was found in many other 2D systems: p-GaAs, n-GaAs, p-Si/SiGe, n-Si/SiGe, n-SOI, p-AlAs/GaAs, etc. Y.Hanein et al. PRL (1998) Papadakis, Shayegan, PRB (1998) n-AlAs-GaAsp-GaAs/AlAs  (  /  )

11. There is no metallic state and no MIT - in the noninteracting 2D systems Spin-orbit interaction ? Electron-phonon interaction ? Too low temperature and too weak e-ph coupling Not renormalized Electron-electron interaction

12. High mobility E ee /E F = r s ~10 density  =4,5m 2 /Vs

13. 13 e-e interaction in Si-MOS structures Note1: high mobility Within the concept of the e-e correlations, the role of high mobility in the 2D MIT becomes transparent The high mobility: Increases  and, hence, the amplitude of interaction corrections (  T  ); Translates down the critical density range (decreases the density of impurities n i ) Increases the magnitude of interaction effects (  F 0   n  ).

14. 2.1. Signatures of the critical phenomenon - QPT Mirror reflection symmetry:  (  n,T)/  c =  c /  (-  n,T) data scaling  /  c = f [T/T 0 (n)] Critical behavior T 0  |n-n c | -z S.V.Kravchenko, W.E.Mason, G.E.Bowker, J.E.Furneaux, V.M.Pudalov, M.D'Iorio, PRB 1995 Symmetry: holds here and is missing outside

15.  =35,000cm 2 /Vs MIT in 2D system (1994)

16.  =35,000cm 2 /Vs MIT in 2D system (1994)

17. 17 2.2. Problems of the data (mis)interpretation If “MIT” is a QPT, it is expected:  c to be universal, scaling persists to the lowest T horizontal “separatrix”  c  f(T) z, are universal Experimentally, however,  c =0.5  5 is sample dependent, z =0.9  2 is sample dependent, reflection symmetry fails at low T and at high T>2K  ins =  c exp(T 0 /T) p1 (p 1 =0.5  1 )  met =  c exp(-T 0 /T) p2 +  0 (p 2 =0.5  1) separatrix is T-dependent The failure of the OPST approach is not surprising: interactions How to proceed in the 2-parameter problem ? Which parameters should be universal ? Definitions of the critical density, critical resistivity etc. ? In analogy with the 1-parameter scaling:

18. 3. Solving the puzzle of the metallic-like conduction at g >>e 2 /h (2000-2004) Ballistic interaction regime T  >>1

19. Quantifying e-e interaction in 2D (2000-2004) F i a,s – FL-constants (harmonics) of the e-e interaction

20. Strong growth in  *  m*g*, m* and g* as n decreases V.M.Pudalov, M.E.Gershenson, H.Kojima, Phys.Rev.Lett. 88, 196404 (2002)

21. Fermi-liquid parameter F 0  N.Klimov, M.Gershenson, VP, et al. PRB 78, 195308 (2008)

22. No parameter comparison of the data and theory in the ballistic regime No parameter comparison of the data and theory in the ballistic regime T  >>1 (2002-2004): Exper.: VP, Gershenson, Kojima, et al. PRL 93 (2004) Theory: Zala, Narozhny, Aleiner, PRB (2001-2002)

23. VP et al. JETP Lett. (1998) Successful description of the transport in terms of e-e interaction effects in the “high density/low disorder (  <<1) regime motivated us to apply the same ideas to the regime of low density/strong disorder (  ~1) 4. Transport in the critical regime

24. Theory: Two- parameter renorm. group equations  is in units of e 2 /h Interplay of disorder and interaction

25. n v =2 Exact RG results for B=0 One-loop, A.A.Finkelstein, Punnoose, Phys.Rev.Lett. (2005)  max

26. Transport data in the critical regime

27. Magnetotransport in the critical regime Quantitative agreement of the data with theory Knyazev, Omelyanovskii, Burmistrov, Pudalov, JETP Lett. (2006) Anissimova, Kravchenko, Punnoose, Finkel'stein, Klapwijk, Nature Phys. 3, 707 (2007) RG equation in B || field: Burmistrov, Chtchelkatchev, JETP Lett. (2006)

28.  2 (T) – comparison with theory Quantitative agreement with theory for both,  (T) and  2 (T)

29. Anissimova, Kravchenko, Punnoose, Finkel'stein, Klapwijk, Nature Phys. 3, 707 (2007)

30. Interplay of disorder and interaction No crossover “2D metal” – localized state No crossover “2D metal” – localized state RG-result in the two-loop approximation Finkelstein, Punnoose, Science (2005)

31. 6. Fixed point (QCP) Two-loop approximation, n v =   c 

32. Data analysis in the vicinity of the fixed point Linearising RG equations close to the fixed point   =   2 = 0:  = p/(2 )  = -py/2 p – for heat capacity, – for correlation length Knyazev, Omelyanovskii, Pudalov, Burmistrov, PRL 100, 046405 (2008)

33. Scaling of the  c (T) data Note: The quality of the data scaling relative the tilted separatrix r c (T) Separatrix – is a power low function, with no maxima and inflection. Exponent  must be < 1. separatrix

34. R(T) data scaling in a wide range of (X,Y >1) Reflection symmetry holds within (0.8%) for |X|<0.5, Y<0.7 Fits 64000 data points to within 4% over the range |X|<5, Y<3 separatrix f 1 = -X+0.07X 2 +0.01X 3 (1-Y+1.48Y 2 ) (1+1.9Y 2 +1.7Y 3 ) f2=f2=

35. Empiric scaling function R(X,Y) – data spline for 5 samples Knyazev, Omelyanovskii, Pudalov, Burmistrov, PRL100, 046405 (2008)

36. Current understanding of the 2D systems  “Metallic” conduction in 2D systems for  >> e 2 /h - the result of e-e interactions  Interplay of disorder and e-e interaction radically changes the common believe that the metallic state can not exist in 2D  Agreement of the data with RG theory and the 2- parameter data scaling M-I-T is a quantum phase transition  In RG theory, the 2D metal always exist for n v =2 (or at large  2 for n v =1), whereas M-I-T is a quantum phase transition Summary More realistic RG calculations are needed (finite n v, two-loop)

37. Thank you for attention! Theory: Sasha Finkelstein - Texas U. Boris Al’tshuler - Columbia U. Igor Aleiner - Columbia U. Dmitrii Maslov - U.of Florida Valentin Kachorovskii - Ioffe Inst. Nikita Averkiev - Ioffe Inst. Alex Punnoose - Lucent Experiment Dima Rinberg - Harvard Univ. Sergei Kravchenko - SEU, Boston, Mary D’Iorio - NRC, Canada John Campbell - NRC, Canada Robert Fletcher - Queens Univ. Gerhard Brunthaler - JKU, Linz Adrian Prinz - JKU, Linz Misha Reznikov - Technion, Haifa Kolya Klimov - Rutgers Univ. Misha Gershenson - Rutgers Univ. Harry Kojima - Rutgers Univ. Nick Busch - Rutgers Univ. Sasha Kuntsevich-Lebedev Inst.