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SOME PUZZLES ABOUT LOGARITHMIC RELAXATION AND A FEW POSSIBLE RESOLUTIONS

SOME PUZZLES ABOUT LOGARITHMIC RELAXATION AND A FEW POSSIBLE RESOLUTIONS. M. Pollak Dept. of Physics, Univ. of CA, Riverside. Introduction – on theory, and an experiment - briefly A question about relaxation to equilibrium. A question about aging theory

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SOME PUZZLES ABOUT LOGARITHMIC RELAXATION AND A FEW POSSIBLE RESOLUTIONS

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  1. SOME PUZZLES ABOUT LOGARITHMIC RELAXATION AND A FEW POSSIBLE RESOLUTIONS M. Pollak Dept. of Physics, Univ. of CA, Riverside • Introduction – on theory, and an experiment - briefly • A question about relaxation to equilibrium. • A question about aging theory • Some needed refinement for relaxation theory Helpful discussions: Amir Frydman Ortuño Ovadyahu Thanks!

  2. 1. Relaxation theory theories for logarithmic relaxation, summary BRIEFLY Pollak and Ovadyahu Phys Stat Sol.C 3, 283, 2006 Amir et. al. PRB 77,165207((2008) The essence: a broad distribution of relaxation processes exp(-wt). w are exponential function of a random variable z in hopping processes z is a combination of energy and hopping distance w~exp(-Eh/kT-r/) Eh is a hopping energy, ra total hopping distance, possibly collective, half the localization length If the distribution n(z) of the random variable zis smooth thenup to logarithmic corrections, n(w)~1/w,n(ln[w])~constant There must exist some cutoff minimal rate wmbelow which n(w) drops off very rapidly.

  3. 10-4 exp(-wt) w=10-12 On a logarithmic plot, exp(-wt) resembles a step function. So E(t) decreases uniformly as the processes gradually decay exp(-wmt) (smallest w) exp(-wt) (sum of future relaxations) t=1/w M. Pollak, M. Ortuño and A. Frydman, The Electron Glass, Cambridge University Press, 2013

  4. log(wm-1)=2log,wm-1=2 Measuring the rate of decay. the two-dip experiment on an MOS structure: gate-insulator-eglass. Protocol (Ovadyahu) observed conductance(Vg,t) evolution of dips Vg2 Vg1 Vg2 dip amplitude of G Same as relaxation experiment Vg1 0 logwm-1 log t time log t0 log many hours traces staggered for clarity G(t)G(t)-G0 Go is measured at this time

  5. log(wm-1)=2log,wm-1=2 2. What can  tell us? Protocol (Ovadyahu) observed conductance(Vg,t) evolution of dips Vg2 Vg1 Vg2 dip amplitude of G Same as relaxation experiment Vg1 0 logwm-1? log t time t01sec log many hours G(t)G(t)-G0 If  should relate to relaxation to equilibrium then G0 must be the equilibriumG. It is often assumed that G measured many hours after cool-down is close to the equilibrium conductance. That may be a mistaken assumption!!!.Equlibrium may not be reached in zillions of years. Grenet and Delahaye, PRB 85, 235114 (2012)

  6. extrapolation from ergodic regime 1020 1018 age of Terra 1016 1014 1012 Quasi ergodic wm (sec) 1010 108 1year time from cool-down? 106 1day 104 102 Arguments that equlibrium G0 may be much smaller than assumed:  Scaling of memory dip with scan rate, If bottom of dip is near equilibrium, such scaling would not be expected.  Calculation of many-electron transition rates. Say that at least 6-e relaxation is needed and r/=4; w-1=0 exp(62r/)=10-12 exp(48)=109sec=O(10years)  Experimental results of  dependence on concentration: below after PRL, 81, 669 (1998) T. Grenet and J. Delahaye, Phys. Rev. B 85, 235114 (2012)

  7. Comments on 2. Can relaxation time to equilibrium be determined? !!! The equilibrium G must be known for that! How to find the equilibrium G? Prepare system in equilibrium? not likelyObtain theoretically? not likely Almost by definition, equilibrium properties of non-ergodic systems cannot be measured. So what is the relevance of experimental  ? It relates to the PAST of the system (e.g. to the time since cool-down) not the FUTURE! T. Grenet and J. Delahaye, Phys. Rev. B 85, 235114 (2012) It can relate to the initial state of the system as prepared. What to study about the e-glass? The connection between the dynamics to history for more complex histories than in the aging experiments Some such studies were already done, Grenet and Delahay, Eur. Phys. J B76,229(2010), Vaknin et. al., PRB 65, 134208 (2002). Relation to the initial state of the system, e.g. preparation at low T (electronic system is at a lower energy) T. Havdala, A. Eisenbach and A. Frydman, EPL 98, 67006 (2012) Generally, relationship between internal state of the system and its dynamics

  8. Comments on 2. Can relaxation time to equilibrium be determined? !!! The equilibrium G must be known for that! How to find the equilibrium G? Prepare system in equilibrium? not likelyObtain theoretically? not likely Almost by definition, equilibrium properties of non-ergodic systems cannot be measured. So what is the relevance of experimental  ? It relates to the PAST of the system (e.g. at high concentration to the time since cool-down) not FUTURE! T. Grenet and J. Delahaye, Phys. Rev. B 85, 235114 (2012) It can relate to the initial state of the system as prepared. What ought one study about the e-glass? The connection between the dynamics to history for more complex histories than in the aging experiments Some such studies were already done, Grenet and Delahay, Eur. Phys. J B76,229(2010), Vaknin et. al., PRB 65, 134208 (2002). Relation to the initial state of the system, e.g. preparation at low T (electronic system is at a lower energy) T. Havdala, A. Eisenbach and A. Frydman, EPL 98, 67006 (2012) A couple more comments: Why should the experimental conductance track VRH theory? (One reason that) it should not: VRH is valid at equilibrium. An argument made against e-glass: critical percolation resistor does not correspond to very long relaxation. Critical resistor has to do with conduction near equilibrium. It can be HUGE.

  9. t=-tw t=0 t 3. Aging There is no standard use of the term. I use it to refer to lack of time homogeneity: startingidenticalexperiments at different times yields different results. Basic reason: non ergodic relaxation, response depends on internal state. Simple experiment: Apply some external force for a time tw and measure response at t>0, (t=0 is start of experiment). Response function black part simulates history, red part is experiment,. t A clear demonstration of time inhomogeneity: the response does not depend on t alone In e-glass the response for such a simple history, (the event at -tw) can be described byf(t/tw) (full aging) Is there a model that can explain time-inhomegeneity and full aging?

  10. 8 ~ ln(1+tw/t) ~ln(1+tw/t) fitted to data at small t/tw 6 G/G (%) 4 D 2 0 -4 -3 -2 -1 0 1 2 3 10 10 10 10 10 10 10 10 t/t w T. Grenet et. al., Eur. Phys. J. B 56, 183 (2007) , and A. Amir, Y. Oreg and Y. Imry, Phys. Rev. Lett. 103, 126403 (2009)more formally, show thatif the path att>0 backtracks exactly (microscopically) the path during 0>t>-tw ,one obtains f(t/tw)=ln(1+tw/t). A very nice agreement with experiment! But a puzzle: Such reversibility implies that sequence of relaxation at t>0 is from slow to fast. A statistical approach yields correct result for t<tw but not for the curved part. M. Pollak, M. Ortuño and A. Frydman, The Electron Glass, Cambridge University Press, 2013 So let’s focus on the curved part!

  11. Consider the same process invoked in the relaxation theory, but restrict the ws to those relaxing during {-tw,0}i.e. replacing wm by1/tw So n(w) decreases sharply at w<1/tw.. Guess an exponential decrease of the random variable z past zm (Poisson distribution) n(z) =C.exp[-a(z-zm)] for z > zm  -ln(wm). (C is an a dependent normalization constant of no importance here.) n(z) exp[-a(z-zm)] wa, (remember w~e-z) n(w) = n(z)(dz/dw) = n(z)/w n(w) wa/w E(t)exp(-wt)n(w)dw =exp(-wt)wa-1dw = t -aexp(-y)ya-1dy The last integral is just an a dependent number, so E(t)t -aatt > tw How does it compare with the other theory ?

  12. Microscopic reversibility vs. Poisson distribution of n(z) ~ln(1+tw/t) ~ t-a A.Vaknin et. al.,PRB 65, 2002 V. Orlyanchik & Z. Ovadyahu, PRL, 92, 066801 (2004) a=0.7 a=0.55 a=0.8 Comments on 3.: Notice that and are very similar for a=0.8. Does full aging extend to t>tw or does relaxation become tw dependent separately? wmtw-1 seems physically more justifiable and in keeping with the relaxation theory.

  13. 10-4 exp(-wt) W=10-12 4. Logarithmic relaxation theory exp(-wmt) exp(-wt)

  14. slow fast e e e The rule that slow decays should follow fast decays has exceptions: After relaxation to a new lower state, a renewal of faster relaxations becomes possible EXAMPLE: spirit of final state

  15. slow (2-electron) decay

  16. e e After a relaxation to a new state, further relaxation to next state can be faster (larger w)

  17. fast (1-electron) decay

  18. e e e slow t fast ghost of initial state This causes relaxation to speed up.On a log time scale it looks like all events with w>1/t that happen after t, happen at t. fast looks like a vertical dropoff on lnt

  19. E relaxation with w=1/t w>1/t relaxations from state at E with probability p(w|E) exp(-wt), w=1/t log t t Is this relaxation still logarithmic?

  20. Is this relaxation still logarithmic? Comments on 4. If p(w|E) is independent of E: relaxation is logarithmic but faster. If p(w|E) is small and the experimental range of t is small compared to {10-12s; wm-1} p(w|E)<<1? As E decreases collective transitions become more dominant.

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