1. Memory effects in electron glasses Markus Müller Eran Lebanon Lev Ioffe Rutgers University, Piscataway NJ 10 August, 2005, Leiden

2. Outline Observation of glassiness in Coulomb systems: Memory and slow relaxation Review of the glass transition Density of states and hopping conductivity Theory of the memory effect, comparison to experiments Summary

3. Glassiness in electronic systems Capacitance: slow relaxation in GaAs, indication for Coulomb gap (D. Monroe et al.) Conductivity: slow relaxation, aging, memory in InO x (Z. Ovadyahu et al.) and in granular Al (T. Grenet) Anomalous noise spectra close to the 2D MIT of Si- MOSFETs (D. Popović et al.) Slow relaxation and hysteresis in granular films (A. Goldman et al., W. Wu et al.) Slow relaxation of phototconductivity in YH 3-   Electron or hydrogen glass? (M. Lee et al.)

4. Electron glasses Unscreened Coulomb interactions M. Pollak (1970) A. Efros, B. Shklovskii (1975) J.H. Davies, P.A. Lee, T.M. Rice (1982,84) C. Yu, T. Vojta, M. Schreiber, A. Möbius etc. Disorder Background charge/gate voltage Anderson insulators with strong electron-electron interactions Occupation of sites on a given lattice Efros-Shklovskii model

5. Electron glasses Unscreened Coulomb interactions Strongly localized electrons → Classical problem with strong frustration Disorder Background charge/gate voltage Long range antiferromagnetic spin glass Anderson insulators with strong electron-electron interactions Occupation of sites on a given lattice Efros-Shklovskii model M. Pollak (1970) A. Efros, B. Shklovskii (1975) J.H. Davies, P.A. Lee, T.M. Rice (1982,84) C. Yu, T. Vojta, M. Schreiber, A. Möbius etc.

6. Glassiness in Capacitance D. Monroe et al., PRL 59, 1148 (1987)

7. Glassiness in Capacitance D. Monroe et al., PRL 59, 1148 (1987)

8. Glassiness in Capacitance Excess length D. Monroe et al., PRL 59, 1148 (1987)

9. Glassiness in Capacitance Excess length D. Monroe et al., PRL 59, 1148 (1987)

10. Glassiness in Capacitance D. Monroe et al., PRL 59, 1148 (1987) Excess length

11. Glassiness in Capacitance D. Monroe et al., PRL 59, 1148 (1987) Excess length Collective glassiness due to frustration or trivially slow dynamics (low T)?

12. Glassiness in Capacitance D. Monroe et al., PRL 59, 1148 (1987) Temperature dependence

13. Electron glass in InOx: M. Ben-Chorin et al., PRL 84, 3402 (2000) Indium-oxides In 2 O 3-x Anomalous field effect

14. Electron glass in InOx: M. Ben-Chorin et al., PRL 84, 3402 (2000) Indium-oxides In 2 O 3-x Anomalous field effect Memory effect A. Vaknin et al., PRL 81, 669 (1998)

15. Glassy behaviour I : Anomalous field effect Granular Au Granular Al Martinez-Arizala et al., PRL 78, 1130 (1997) T. Grenet, EPJ B 32, 275 (2003) Granular Pb Adkins et al., JPC 17, 4633 (1984) M. Ben-Chorin et al., PRL 84, 3402 (2000)

16. A. Vaknin et al., PRL 84, 3402 (2000) Indium-oxides In 2 O 3-x Electron glass in InOx: Relaxation and aging

17. A. Vaknin et al., PRL 84, 3402 (2000) Simple aging! Indium-oxides In 2 O 3-x Electron glass in InOx : Relaxation and aging

18. A. Vaknin, Z. Ovadyahu, and M. Pollak, PRL 81, 669 (1998) Protocol: Imprint gate voltage V g = 0 Change V g to new value Sweep V g from time to time, measure conductivity G(V g ) Electron glass in InOx : Memory effect

19. Z. Ovadyahu, Phil. Mag. B 81, 1225 (2001) Properties of the memory cusp The normalized shape of the anomalous field effect is independent of disorder strength and magnetic field (localisation length)!

20. Z. Ovadyahu, Phil. Mag. B 81, 1225 (2001) Properties of the memory cusp The normalized shape of the anomalous field effect is independent of disorder strength and magnetic field (localisation length)! widens with increasing carrier density (evidence for Coulomb interaction!)

21. Z. Ovadyahu, Phil. Mag. B 81, 1225 (2001) Properties of the memory cusp The normalized shape of the anomalous field effect is independent of disorder strength and magnetic field (localisation length)! widens with increasing carrier density (evidence for Coulomb interaction!) becomes narrower and deeper with decreasing temperature Why ? Relation to the Coulomb gap?

22. Summary of experiments Out of equilibrium behaviour of electron glasses: Very slow capacitance relaxation Slow relaxation, aging and memory effect in conductivity Clear signatures of electron-electron interactions Quantitative theory of the anomalous field effect? Relation between conductivity cusp and Coulomb gap? ??

23. Range of validity of the ES-model Nearest neighbor interaction Strong disorder limit  Level spacing in the localisation volume Efros-Shklovskii model applicable!

24. Density of states and hopping Local density of states in the classical (strongly localized) limit where(Local fields) Efros-Shklovskii: Universal! (Due to marginality of the glass)

25. Density of states and hopping Local density of states in the classical (strongly localized) limit where(Local fields) Efros-Shklovskii: Universal! (Due to marginality of the glass) Hopping: Optimize transition probability High T: Mott variable range hopping Low T: Efros-Shklovskii hopping

26. Analogous Gap in long ranged spin glasses: Local fields Thouless, Anderson and Palmer, (1977) Palmer and Pond, (1979) Bray and Moore, (1980) Sommers and Dupont, (1984) with

27. Coulomb gap : DC conductivity J. G. Massey and M. Lee, PRL 87, 056402 (2001) Mott Efros-Shklovskii Mott conductivity : Constant density of states Hopping: Optimize

28. Coulomb gap : DC conductivity Mott conductivity : Constant density of states Efros-Shklovskii conductivity: Coulomb gap J. G. Massey and M. Lee, PRL 87, 056402 (2001) Hopping: Optimize Mott Efros-Shklovskii

29. The Glass transition 3D 2D Weak disorder (Low density InO x, standard doped semiconductors) Strong disorder (High density InO x ) C.Yu et al. (93) S. Pankov and V. Dobrosavljevic (04) M. Müller and L. Ioffe (04) MF

30. The Glass transition 3D 2D Weak disorder (Low density InO x, standard doped semiconductors) Strong disorder (High density InO x ) C.Yu et al. (93) S. Pankov and V. Dobrosavljevic (04) M. Müller and L. Ioffe (04) DOS MF

31. The Glass transition 3D 2D Weak disorder (Low density InO x, standard doped semiconductors) Strong disorder (High density InO x ) C.Yu et al. (93) S. Pankov and V. Dobrosavljevic (04) M. Müller and L. Ioffe (04) DOS Hardly observable! MF

32. The Glass transition 3D 2D Weak disorder (Low density InO x, standard doped semiconductors) Strong disorder (High density InO x ) C.Yu et al. (93) S. Pankov and V. Dobrosavljevic (04) M. Müller and L. Ioffe (04) DOS Mott conductivity at T c ! MF

33. From 3D to 2D: Thin films Glass transition

34. From 3D to 2D: Thin films Glass transition Hopping conductivity

35. From 3D to 2D: Thin films Glass transition Hopping conductivity Experiments on memory effect

36. A phenomenological theory of the memory cusp

37. 2D Density of states Density of states out of equilibrium T = 0 V = 0

38. 2D Density of states Density of states out of equilibrium T = 0 V = 0 T finite V = 0

39. 2D Density of states Density of states out of equilibrium T = 0 V = 0 T finite V = 0 T finite V finite

40. Numerical evidence for shifted DOS at T=0 Procedure Find metastable at T = 0. Increase gate voltage (background charge density  ) ↔ Inject new particles with density  Steepest descent to new metastable state Measure new density of states with respect to new E F.

41. Numerical evidence for shifted DOS at T=0 Procedure Find metastable at T = 0. Increase gate voltage (background charge density  ) ↔ Inject new particles with density  Steepest descent to new metastable state Measure new density of states with respect to new E F.

42. Numerical evidence for shifted DOS at T=0 Procedure Find metastable at T = 0. Increase gate voltage (background charge density  ) ↔ Inject new particles with density  Steepest descent to new metastable state Measure new density of states with respect to new E F.

43. Numerical evidence for shifted DOS at T=0 Procedure Find metastable at T = 0. Increase gate voltage (background charge density  ) ↔ Inject new particles with density  Steepest descent to new metastable state Measure new density of states with respect to new E F.

44. Numerical evidence for shifted DOS at T=0 Procedure Find metastable at T = 0. Increase gate voltage (background charge density  ) ↔ Inject new particles with density  Steepest descent to new metastable state Measure new density of states with respect to new E F.

45. Numerical evidence for shifted DOS at T=0 Procedure Find metastable at T = 0. Increase gate voltage (background charge density  ) ↔ Inject new particles with density  Steepest descent to new metastable state Measure new density of states with respect to new E F.

46. Numerical evidence for shifted DOS at T=0 Procedure Find metastable at T = 0. Increase gate voltage (background charge density  ) ↔ Inject new particles with density  Steepest descent to new metastable state Measure new density of states with respect to new E F.

47. Numerical evidence for shifted DOS at T=0 Procedure Find metastable at T = 0. Increase gate voltage (background charge density  ) ↔ Inject new particles with density  Steepest descent to new metastable state Measure new density of states with respect to new E F.

48. Finite T: Instability and the width of the cusp Instability of the glass state:

49. Finite T: Instability and the width of the cusp Instability of the glass state: Beyond V crit the reached state is unstable and runs downhill (but reversibly) to the closest metastable state. Saturation of the out of equilibrium effect.

50. Finite T: Instability and the width of the cusp V crit as a function of temperature Instability of the glass state: Experiment V crit as a function of carrier density

51. Finite T: Instability and the width of the cusp V crit as a function of temperature Instability of the glass state: Experiment V crit as a function of carrier density

52. Finite T: Instability and the width of the cusp V crit as a function of temperature Instability of the glass state: Experiment V crit as a function of carrier density

53. Percolation picture for hopping conductivity Groundstate configuration occupied empty

54. Percolation picture for hopping conductivity Groundstate configuration occupied empty Bonding criterion

55. Percolation picture for hopping conductivity Groundstate configuration occupied empty Bonding criterion

56. Percolation picture for hopping conductivity Groundstate configuration occupied empty Bonding criterion Percolation! →  crit ~ R(T) R(T) as a functional of the non-equilibrium DOS!

57. Resistivity as a function of T Crossover from Mott to Efros-Shklovskii

58. Cusp shape as a function of T Experiment Theory

59. Cusp amplitude Experiment Theory A. Vaknin and Z. Ovadyahu, (1998)

60. Cusp amplitude Experiment Theory A. Vaknin and Z. Ovadyahu, (1998)

61. Effect of disorder annealing and magnetic field Assumption: Only  affected Only small effect on relative amplitude! Strong effect on absolute amplitude!

62. Summary Phenomenological theory for the anomalous field effect (memory cusp) in electron glasses: The cusp results from the rigidity of the glass against collective rearrangements The glassy state becomes unstable when more particles are injected than are thermally active → saturation of the cusp at V crit → decreasing cusp width and increasing amplitude with decreasing T → low sensitivity to disorder and magnetic field

63. M. Lee et al., PRB 60, 1582 (1999) From Coulomb gap to zero bias anomaly Quantum melting of the electron glass