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Glassy dynamics near the two-dimensional

Glassy dynamics near the two-dimensional metal-insulator transition. J. Jaroszy ński and Dragana Popovi ć National High Magnetic Field Laboratory Florida State University, Tallahassee, FL. Acknowledgments:

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Glassy dynamics near the two-dimensional

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  1. Glassy dynamics near the two-dimensional metal-insulator transition J. Jaroszyński and Dragana Popović National High Magnetic Field Laboratory Florida State University, Tallahassee, FL Acknowledgments: NSF grants DMR-0071668, DMR-0403491; IBM, NHMFL; V. Dobrosavljević, I. Raičević

  2. Si MOSFET Si Background • metal-insulator transition (MIT) in 2D electron and hole systems • in semiconductor heterostructures (Si, GaAs/AlGaAs, …) • critical resistivity ~h/e2 • role of disorder? • rs  U/EF  ns-1/2  10 • role of Coulomb interactions?

  3. competition between disorder and Coulomb interactions: • glassy ordering??? • [Davies, Lee, Rice, PRL 49, 758 (1982); • 2D: Chakravarty et al., Philos. Mag. B 79, 859 (1999); Thakur et al., • PRB 54, 7674 (1996) and 59, R5280 (1999); Pastor, Dobrosavljević, • PRL 83, 4642 (1999)] • Ourearlier work: transport and resistance noisemeasurements • to probe the electron dynamics in a 2D system in Si MOSFETs •  signatures of glassy dynamics in noise • 2D MIT in Si: melting of the Coulomb glass

  4. -Metal: (T=0)0; -Slow, correlated dynamics (1/f noise;   1.8) (ns,T)=(ns,T=0)+b(ns)T3/2 nc ng ns* density [Bogdanovich, Popović, PRL 88, 236401 (2002); Jaroszyński, Popović, Klapwijk, PRL 89, 276401 (2002); Jaroszyński, Popović, Klapwijk, PRL 92, 226403 (2004)] T=0 phase diagram • -Metal: (T=0)0; • d/dT<0 • Fast, uncorrelated • dynamics • (1/f noise; =1) -Insulator: (T=0)=0 -Slow, correlated dynamics (1/f  noise;   1.8) ns*– separatrix (from transport) ng – onset of slow dynamics (from noise) nc – critical density for the MIT from (T) on both insulating and metallic sides High disorder(low-mobility devices):nc < ng < ns* Low disorder(high-mobility devices):nc  ns* ≲ ng for B=0, nc < ns* ≲ ng for B≠0 (Coulomb glass) Theory: Dobrosavljević et al.

  5. slow relaxations and history dependence of (ns,T) also observed • for ns < ng This work: a systematic study of relaxations as a function of ns and T • Samples: • low-mobility (high disorder) Si MOSFETs with LxW of 2x50 and 1x90 m2 • [from the same wafer as those used for noise measurements in • Bogdanovich et al., PRL 88, 236401 (2002); all samples very similar] • data presented for 2x50 m2 sample • Note: critical densitync(1011cm-2)  4.5 obtained from (ns,T=0) in • (ns,T)=(ns,T=0) + b(ns)T3/2, • which holds slightly above nc (up to n  0.2); below nc,  is insulating • (decreases exponentially with decreasing T) - similar to published data • noise measurements in this sample give ng(1011cm-2)  7.5, the same as • published results

  6. Example 1: Sample annealed @ Vg=11V (ns=20.26 x 1011cm-2) @ T=10K; then cooled down to different T (here to 3.5 K); then @ t = 0, Vg switched (here) to Vg=7.4 V (ns=4.74 x 1011cm-2) and relaxation measured. After change of Vg,  decreases fast, goes through a minimum and then relaxes up towards 0,which is  when sample is cooled down at Vg=7.4 V (i.e. equilibrium ). To measure 0, after some time (here approx. 55000 s), T is increased up to 10 K to rejuvenate the sample and then lowered back to 3.5 K. Note: large perturbation

  7. Example 2: Sample annealed @ Vg=11V @ T=10K; then cooled down to different T (here to 1 K); then @ t = 0, Vg switched (here) to Vg=7.4 V and relaxation measured. After change of Vg,  initially decreases fast to below 0, and then continues to decrease slowly. In both cases, the system first moves away from equilibrium.

  8. Relaxations at different temperatures for a fixed final Vg=7.4 V

  9. 1.2 K 3.7 K 2.4 K 4.4 K 3.2 K I “Short” t (i.e. just before the minimum in ): data collapse as shown after a horizontal shift low(T)anda vertical shift a(T). This means scaling: /0=a(T)g(t/low(T)) a(T)  (low)- Scaling function: linear on a ln [/(0a(T))] vs. (t/low)(=0.3 for Vg=7.4V) scale for over 4 orders of magnitude in t/low,i.e. a stretched exponential dependence for intermediate times(just below minimum in (T)). [ a(T))] Vg=7.4 V

  10. At even shorter times(best observable at lowest T): power-law dependence/0  t- (dashed lines are linear least squares fits with slopes 0.068 at 0.4 K and 0.071 at 0.3 K ) In this region, scaling may be achieved by a nonunique combination of horizontal and vertical shifts.

  11. Scaling: [ a(T)] Vg=7.4 V /low(T) At lowest T (< 1.2 K), stretched exponential crosses over to a power law dependencewith an exponent 0.07 but scaling in the power law region is not unambiguous.

  12. 0.24 K 0.5 1.0 4.4 2.4 Vg=7.4 V 3.2 Can we describe all the data with the following (Ogielski) scaling function? /0  t- exp[-(t/low)] = (low)- (t/low)- exp[-(t/low)] f(t/low) (It works in spin glasses: C. Pappas et al., PRB 68, 054431 (2003) in Au0.86Fe0.14) Yes! black dashed line – fit to Ogielski form

  13. 1.2 K 2.4 4.4 3.2 A blowup of the region where curves collapse well: black dashed line – fit to Ogielski form Vg=7.4 V • curves collapse well down to 0.8 – 1.2 K; extract exponents  and  • experiment and analysis repeated for different Vg, i.e. ns: relaxations • measured after a rapid change of Vg from 11 V to a given Vg at many • different T

  14. individual fits Ogielski formula exponent  individual fits Ogielski formula  •  -power law exponent; •  - stretched exponential • exponent • dashed lines are guides • to the eye • nc (1011cm-2)  4.5 ns (1011 cm-2) • →0 at ns(1011cm-2)  7.5-8.0  ng,where ng • was obtained from noise measurements!!! •  grows with ns – relaxations faster

  15. Scaling parametersvs. T log[low(2.4 K)/low(T)] vs. 1/T • black line is an Arrhenius fit to the data in the regime where curves collapse well;Arrhenius fitworks well over7 orders of magnitude in 1/low

  16. 1/low(T)= k0 exp(-Ea/T),with Ea19 K and k0  6.25 s-1 • for Vg=7.4 V • similar results are obtained for other Vg in the glassy region • (e.g. Ea20.8 K for Vg=7.2 V, and Ea22 K for Vg=8.0 V) •  Ea  20 K, independent of Vg in this range • (3.99 ≤ ns(1011cm-2) ≤ 7.43; 29 ≤ EF (K) ≤ 54) • but k0=k0(Vg), i.e. k0=k0(ns)

  17. low (s) individual fits Ogielski formula  ns (1011 cm-2) 1/low(ns,T)= k0(ns) exp(-Ea/T) • dashed lines • guide the eye T=3 K • a decrease of low with decreasing ns does not imply that the system is faster • at low ns; since the dominant effect is the decrease of , the system is actually • slower at low ns

  18. ln low (s) [ns (1011 cm-2)]1/2 ns (1011 cm-2) ln low (s) Blue line – fit to ns1/2 low(T) exp(ans1/2) exp(Ea/T), Coulomb energy U  ns1/2; 1/rs=EF/U ~ ns1/2 • strong evidence for the dominant role of Coulomb interactions between 2D • electrons in the observed slow dynamics

  19. Vg=7.4 V II Long t (i.e. above minimum in (t), observable at highest T): all collapse onto one curve after horizontal shift(no vertical shift needed, as expected: all relax to 0 i.e. to 0 on this scale). Data collapsed onto T=5 K curve. This means scaling: /0= f(t/high(T))

  20. Vg=7.4 V (0-)/0exp(-k1t/high) (0-)/0exp(-k2t/high) Scaling function– describes relaxation of  to 0 from below. There are two simple exponential regions (the slower one is not always seen).

  21. Scaling parameter highvs. T • high  exp[EA/(T-T0)], • T0  0, EA  57 K (Arrhenius) Vg=7.4 V

  22. Characteristic times high/k1 and high/k2do not depend on Vg in the range shown; they also do not depend on the direction of Vg change (see below).The data shown were obtained by changing Vg between the values given on the plot. The fits on this plot were made to all points. Final Vgs (7.2 to 11) correspond to a density range from 3.99 to 20.36 in units of 1011cm-2 (EF from 29 K to 149 K). • highexp[EA/(T-T0)], • T0=0, EA  57 K

  23. Conclusions • the system appears glassy for short enough t < (high/k1) : • relaxations have the Ogielski form  t- exp[-(t/low)], • with low  exp (ans1/2) exp (Ea/T), Ea20 K • the system reaches equilibrium at (high/k1)<(high/k2) << t: • relaxations exponential (high  exp (EA/T), EA 57 K) high→  as T→0, i.e.Tg = 0 Examples of time scales: T=5 K, high/k1  34 s; T= 1 K, high/k1  1013 years! (age of the Universe  1010 years) [see Grempel, Europhys. Lett. 66, 854 (2004)] • consistent with noise measurements Note: The system reaches equilibrium only after it first goes farther away from equilibrium! [Also observed in orientational glasses and spin glasses; see also “roundabout” relaxation: Morita and Kaneko, PRL 94, 087203 (2005)]

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