Nonequilibrium dynamics in Coulomb glasses near the metal-insulator transition

Transcription

1 Nonequilibrium dynamics in Coulomb glasses near the metal-insulator transition Dragana Popović National High Magnetic Field Laboratory Florida State University, Tallahassee, FL, USA Boulder 2009 Summer School Supported by NSF DMR and NHMFL (NSF and the State of Florida)

2 Lecture I: Metal-insulator transition and complexity in electronic systems Lecture II: Studies of the electron dynamics near the 2D MIT: Relaxations of conductivity 2D electron system in Si: temperature dependence of conductivity Relaxations of conductivity after a rapid change of density Relaxations after a waiting time protocol: aging and memory Aging across the 2D MIT Lecture III: Studies of the electron dynamics near the 2D MIT: Fluctuations of conductivity

3 Studies of the electron dynamics near the 2D MIT Problem: strongly interacting electrons in a random potential 2D electron system in Si MOSFETs; samples with very different amounts of disorder σ (Electric) field effect: conductivity σ(v g ) Drude σ=n s eμ V g peak mobility at 4.2 K rough measure of disorder vary density n s using V g Evidence for phase transition(s)? MIT, glass transition?

4 Resistivity in high-mobility (low disorder) Si MOSFETs high-mobility (~ 25,000 cm 2 /Vs) Si MOSFETs (Groningen/Delft); L=120 μm, W=50 μm (d 0x =147 nm, Al gates, N a ~10 14 cm -3 ) separatrix n s * lowest n s and T: <ρ> exp(e a /k B T) n c cm -2 n c n s * critical density (where strong localization ends) n c n g glass transition Will show this later

5 Conductivity in low-mobility (high disorder) Si MOSFETs low-mobility (~600 cm 2 /Vs) Si MOSFETs (IBM); L W: 1x90 and 2x50 μm 2 (d ox =50 nm, poly-si gates, N a ~10 17 cm -3 ) metallic <σ(t)> at high n s d<σ>/dt=0 at n s* = cm -2 ( separatrix ) metal-insulator transition: n c =(5.0±0.3) cm -2 glass transition (will show later): n g =(7.5±0.3) cm -2 < > -time average metallic glass k F l < 1 ( bad metal)

6 [S. Bogdanovich and D. Popović, PRL 88, (2002)] at the lowest n s, strongly localized: <σ> exp(-t o /T), n c =(5.0±0.3) cm -2, n c n s * just above n c (metallic glass): <σ>=a(n s ) + b(n s )T x, x 1.5 non-fermi liquid behavior (not a good metal) <σ(n c,t)> T 3/2 (a power law, as it should be for the MIT) (consistent with V. Dalidovich and V. Dobrosavljević, PRB 66, (2002), for the metallic glass phase)

7 Back to high-mobility samples; apply parallel magnetic field B B 2DES spin-polarized (no orbital effect; B couples only to spins) Bad (NFL) metal Insulator [Jaroszyński, Popović, Klapwijk, PRL 92, (2004)] B=0: (almost) no intermediate phase Apply B: emergence of intermediate phase with the same σ(t) as in samples with high disorder (suppression of screening by parallel B effective disorder increases)

8 Intermediate metallic phase σ(n s,t,b)=σ(n s,t=0,b)+b(n s,b)t 3/2! (same as in low-mobility samples at B=0!) σ(n s,t=0,b) δ nμ, μ 1.5 consistent with QPT n s (10 10 cm -2 )=11.9, 11.6, 11.3, 11.2, 11.0, 10.9, 10.7 from top; n c (B=2T)= cm -2 at B=0, μ [Fletcher et al., Semicond. Sci. Tech. 16, 386 (2001)]

9 T=0 phase diagram Intermediate phase! Insulator: σ(t=0)=0 Strange metal: σ(t=0) 0 σ(n s,t)=σ(n s,t=0)+b(n s )T 3/2 Metal: σ(t=0) 0; dσ/dt<0 will show this is a n s* separatrix (from transport) glass transition n g glass transition (will show later) critical density for the MIT from σ(t) on both insulating and metallic sides n c High disorder (low-mobility devices): n c < n g < n s * Low disorder (high-mobility devices): n c n s * n g for B=0, n c < n s * n g for B 0 n c n g n s * density [Bogdanovich, Popović, PRL 88, (2002); Jaroszyński, Popović, Klapwijk, PRL 89, (2002); Jaroszyński, Popović, Klapwijk, PRL 92, (2004)]

10 How to probe glassy dynamics? measure response of the system to some kind of a perturbation (e.g. after a rapid cooling; a spin glass in a magnetic field) here, perturbation = change of V g ; measure conductivity σ vs. time t after the perturbation is switched off supercooled water [see also papers by Z. Ovadyahu for similar work in InO x electron glass deep in the insulating regime]

11 Relaxations of conductivity after a rapid change of n s n g cm -2, n c cm Low-mobility samples Initial n s (10 11 cm -2 ) =20.26 > n g k F l 1 σ (e 2 /h) Vg(V) T (K) Overshooting of equilibrium! σ T=3.3 K t/s Time σ 0 equilibrium conductivity at T and final n s Final n s (10 11 cm -2 )= =4.74 n c ΔE F»k B T [J. Jaroszyński and D. Popović, PRL 96, (2006)]

12 Repeat measurement at (many) different T (after warm-up to 10 K): minimum moves to longer times as T decreases slower relaxations

13 Approach to equilibrium: data (for different T) collapse for times after the minimum Relaxations exponential (τ high τ eq ) Characteristic (equilibration) time τ eq exp (E A /T), The system reaches equilibrium after a long enough t E A 57 K τ 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!]

14 Initial relaxation: for short enough t <τ eq, data (for different T) collapse for times before the minimum: σ(t,t)/σ 0 t -α(n) exp{-[t/τ low (n s,t)] β(n) } (n n s ) glassy relaxation (α=0.07, β<0.3 for this n s ) τ low f(n s ) exp (E a /T), E a 20 K

15 Repeat everything for many different n s τ low exp (an s 1/2 ) exp (E a /T), E a 20 K T 0: σ/σ 0 t -α(n) as expected for a phase transition at T=0 (previous slide: scaling as T 0) α(n s ) 0 as n s n g (no slow relaxation for n s > n g ) Coulomb interactions in 2D: E F /U ~ n s 1/2

16 What have we learned from relaxations? data strongly suggest T g =0 for n s n g in a 2DES in Si (diverging equilibration time, scaling of nonexponential relaxations, power law as T 0 T g = 0; similar behavior in spin glasses, where T g 0) at finite T, the system appears glassy for short enough t (e.g. at T= 1 K, equilibration time years! age of the Universe years) Coulomb interactions between 2D electrons a dominant role in the out-of-equilibrium dynamics as T 0, no relaxations for n s > n g ; no relaxations for k F l > 1 Note: system equilibrates only after it first goes farther away from equilibrium!

17 Phase diagram of a 2DES in Si Temperature (K) 0 Insulating σ(t 0)=0 Glassy Behavior (for n s <n g ) dσ/dt>0 k F l < 1 Metallic (Non-Fermi Liquid) σ(t 0) 0 n c intermediate, glassy phase n g dσ/dt<0 Metallic (FL? NFL?) n s * Glassy regime: slow, nonexponential relaxations, diverging equilibration time (T g =0)

18 Times 0 t t=0 preparation time t w waiting time In equilibrium τ eq < t m Out of equilibrium τ eq > t m t m = t +t w measurement time time Out of equilibrium: responses and correlations depend on two times, t and t w (age) - aging Slow Relaxations and Nonequilibrium Dynamics in Condensed Matter, edited by J.-L. Barrat, M.V. Feigelman, J. Kurchan, J. Dalibard (Springer, New York, 2003) - Les Houches summer school Ageing and the Glass Transition, edited by M. Henkel, M. Pleimling, R. Sanctuary (Lecture Notes in Physics, Springer, 2007) Univ. of Luxemburg summer school

19 Relaxations of conductivity after a waiting time protocol: aging and memory 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 [J. Jaroszyński and D. Popović, Phys. Rev. Lett. 99, (2007)]

20 Relaxations for a few different T and t w : overshooting Memory 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.

21 And a few more : Memory loss

22 When is overshooting observed? 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

23 What is the origin of overshooting??? observed in a variety of systems (e.g. insulating granular metals, manganites, biological systems) some theoretical models [Morita et al., PRL 94, (2005); Mauro et al., PRL 102, (2009)] large perturbations out of equilibrium? here ΔE F >> T should trigger major charge rearrangements (n s changed up to a factor of 7; in InO x, density change ~ 1%)

24 Remove all 2D electrons from the inversion layer during t w (V 1 <V T ): t w (s): T=1 K log σ(t,t)/σ 0 (T) log t (s) No t w dependence, i.e. no memory! Glassiness from 2DES, not from background charges

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

26 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 wμ ) (μ-scaling useful in studies of other glasses; may not have a clear physical meaning)

27 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 wμ ) (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 ) NOTE: mean-field models of glasses include both those that show full aging and those where no t/t w scaling is expected. insulating glassy phase and metallic glassy phase are different!

28 μ does not depend on temperature two different samples

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

30 Summary of Lecture II Emergence of an intermediate, (NFL) metallic phase (n c < n g ) between the metal and the insulator Glassy behavior for n s < n g (in the insulator and in the intermediate phase) glassy ordering as a precursor of the MIT in a 2DES in Si abrupt changes in aging at the MIT 2DES in Si: - similarities to other glassy systems (e.g. spin glasses) -a simple, model system for exploring the dynamics of strongly correlated systems (free of complications associated with changes in magnetic or structural symmetry) Lecture III: other probes of the electron dynamics fluctuations of σ