1. Glassy dynamics of electrons near the metal-insulator transition in two dimensions Acknowledgments: NSF DMR-0071668, DMR-0403491, NHMFL; IBM-samples; V. Dobrosavljević – discussions Dragana Popović National High Magnetic Field Laboratory Florida State University, Tallahassee, FL, USA S. Bogdanovich, NHMFL/FSU (now at Hologic Inc.) J. Jaroszyński, NHMFL/FSU T. M. Klapwijk, Delft Univ. of Tech. Collaborators:

2. Glassiness in electronic systems Two-dimensional (2D) electron system in Si: general problem: strongly interacting electrons in a random potential 2D metal-insulator transition (MIT) Studies of the electron dynamics near the 2D MIT: Resistance noise and relaxations in a 2DES in Si Evidence for glassiness: slow, correlated dynamics nonexponential relaxations diverging equilibration times as T→0 aging and memory Conclusions: 2D MIT- melting of a Coulomb glass 2DES in Si: an excellent model system for studying glassy dynamics Outline

3. Glassiness in electronic systems expected in Anderson insulators with strong electron-electron interactions [M. Pollak (1970); A. Efros, B. Shklovskii (1975); J. H. Davies, P. A. Lee, T. M. Rice (1982,84)] competition between Coulomb interactions and disorder Frustration! FluidGlass configuration space many metastable states with similar (free) energy few experimental studies

4. 2D metal-insulator transition Si SiO 2 [Kravchenko et al., PRB 51, 7038 (1995)] Resistivity Temperature critical resistivity ~h/e 2 ; k F l ~ 1 strong disorder r s  U/E F  n s -1/2  10 strong Coulomb interactions n s * - “separatrix” (n c ?–critical density for the MIT; NO!) 2D electrons and holes in semiconductor heterostructures at low densities n s ; our work: 2DES in Si MOSFETs

5. Studies of the electron dynamics near the 2D MIT Problem: strongly interacting electrons in a random potential Our approach: study insulator and electron dynamics as density is varied through the MIT Evidence of a phase transition? Experiments: noise spectroscopy (we measure resistance noise) relaxations of conductivity use samples with very different amounts of disorder

6. Temperature (K) ns*ns* 0 d  /dT<0d  /dT>0 Glassy Behavior (for n s <n g ) Insulating  (T  0)=0 Metallic (Non-Fermi Liquid)  (T  0)≠0 k F l < 1 Metallic (FL? NFL?) ncnc ngng Phase diagram of a 2DES in Si [Bogdanovich, Popović, PRL 88, 236401 (2002); Jaroszyński, Popović, Klapwijk, PRL 89, 276401 (2002); Jaroszyński, Popović, Klapwijk, PRL 92, 226403 (2004)] Special case of low disorder: n c  n s * ≲ n g for B=0, (no intermediate phase) n c < n s * ≲ n g for B≠0 (emergence of intermediate phase) glassy ordering due to charge, not spin intermediate, glassy phase Glassy regime: slow, correlated dynamics; Other manifestations of glassiness?

7. Relaxations of conductivity after a waiting time protocol: aging and memory [J. Jaroszyński and D. Popović, Phys. Rev. Lett. 99, 046405 (2007)] Initial and final n s (10 11 cm -2 )=3.88 < n c ; density during t w =1000 s: n s (10 11 cm -2 )=20.26 > n g change history by varying T and t w

8. Relaxations for a few different T and t w : Response (conductivity) depends on the system history (t w and T) in addition to the time t – aging – a key characteristic of relaxing glassy systems. Memory overshooting

9. overshooting only when the system is excited out of a thermal equilibrium (t w » τ eq ); no memory no OS when excited out of a relaxing (nonequil.) state (t w « τ eq ): aging and memory Equilibration time τ eq  exp(E A /T), E A  57 K  eq →  as T→0, i.e. glass transition T g = 0 [see Grempel, Europhys. Lett. 66, 854 (2004) for a 2D Coulomb glass; also showed aging!] [J. Jaroszyński and D. Popović, PRL 96, 037403 (2006)]

10. Aging regime (no OS, T=1 K) IV [J. Jaroszyński and D. Popović, Phys. Rev. Lett. 99, 216401 (2007)] (T= 1 K: τ eq  10 13 years! Age of the Universe  10 10 years) n 0 < n c Full (simple) aging: σ(t/t w ) σ(t)/σ 0  (t/t w ) -  for t ≤ t w  a memory of t w is imprinted on each σ(t)

11. σ (t, t w ) exhibit full aging for n s < n c for n s > n c, an increasingly strong departure from full aging that reaches maximum at n g aging function: σ (t/t w μ ) (μ-scaling useful in studies of other glasses; may not have a clear physical meaning)

12. σ (t, t w ) exhibit full aging for n s < n c for n s > n c, an increasingly strong departure from full aging that reaches maximum at n g aging function: σ (t/t w μ ) (μ-scaling useful in studies of other glasses; may not have a clear physical meaning) full aging: μ=1 an abrupt change in aging at the 2D MIT (n c ) insulating glassy phase and metallic glassy phase are different! NOTE: mean-field models of glasses include both those that show full aging and those where no t/t w scaling is expected.

13. σ(t)/σ 0 =[σ(t=1s)/σ 0 ] t -  Relaxation amplitudes peak just below n c, and they are suppressed in the insulating regime! both relaxation amplitudes σ(t=1s)/σ 0 and slopes  depend nonmonotonically on n 0 another change in aging properties at n s  n c ncnc ngng Fixed t w and n 1 ; vary n 0

14. Conclusions 2DES in Si exhibits aging and memory – hallmarks of glassy behavior [PRL 99, 046405 (2007)] abrupt changes in the 2DES dynamics at the MIT and at n g [PRL 99, 216401 (2007)] 2D MIT in a 2DES in Si: melting of the Coulomb glass (regardless of the device size, geometry, and fabrication details) similarities to other glassy systems: slow, correlated dynamics, nonexponential relaxations, diverging equilibration times, aging, memory 2DES in Si an excellent model system for exploring the dynamics of strongly correlated systems (free of “complications” associated with changes in magnetic or structural symmetry) I II