SOME PUZZLES ABOUT LOGARITHMIC RELAXATION AND A FEW POSSIBLE RESOLUTIONS M. Pollak Dept. of Physics, Univ. of CA, Riverside 1.Introduction – on theory, and an experiment - briefly 2.A question about relaxation to equilibrium. 3.A question about aging theory 4.Some needed refinement for relaxation theory Helpful discussions: Amir Frydman Ortuño Ovadyahu Thanks!

theories for logarithmic relaxation, summary BRIEFLY 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(-E h /kT-r/  ) E h is a hopping energy, r a total hopping distance, possibly collective,  half the localization length If the distribution n(z) of the random variable z is smooth then up to logarithmic corrections, n(w)~1/w, n(ln[w])~constant 1. Relaxation theory There must exist some cutoff minimal rate w m below which n(w) drops off very rapidly. Pollak and Ovadyahu Phys Stat Sol.C 3, 283, 2006 Amir et. al. PRB 77,165207((2008)

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

Measuring the rate of decay. the two-dip experiment on an MOS structure: gate-insulator-eglass. Vg1Vg1 Vg2Vg2 time log t dip amplitude of  G log  Vg2Vg2 Vg1Vg1 Protocol (Ovadyahu) observed conductance(V g,t) evolution of dips Same as relaxation experiment logw m -1 0 log(w m -1 )=2log , w m -1 =  2  G(t)  G(t)-G 0 many hours G o is measured at this time log t 0 traces staggered for clarity

2. What can  tell us? If  should relate to relaxation to equilibrium then G 0 must be the equilibrium G. Vg1Vg1 Vg2Vg2 time log t dip amplitude of  G log  Vg2Vg2 Vg1Vg1 Protocol (Ovadyahu) observed conductance(V g,t) evolution of dips Same as relaxation experiment logw m -1 ? 0 log(w m -1 )=2log , w m -1 =  2 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.  G(t)  G(t)-G 0 t 0  1sec many hours Grenet and Delahaye, PRB 85, (2012)

Arguments that equlibrium G 0 may be much smaller than assumed:  Scaling of memory dip with scan rate,  Calculation of many-electron transition rates. Say that at least 6-e relaxation is needed and r/  =4; w -1 =  0  exp(6  2r/  )=  exp(48)=10 9 sec=O(10years)  Experimental results of  dependence on concentration: below Quasi ergodic extrapolation from ergodic regime w m (sec) day 1year age of Terra time from cool-down? after PRL, 81, 669 (1998) If bottom of dip is near equilibrium, such scaling would not be expected. T. Grenet and J. Delahaye, Phys. Rev. B 85, (2012)

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 likely Obtain 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, (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, (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, (2012) Generally, relationship between internal state of the system and its dynamics Comments on 2.

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 H U G E. 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 likely Obtain 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, (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, (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, (2012)

3. Aging There is no standard use of the term. I use it to refer to lack of time homogeneity: starting identical experiments 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 t w and measure response at t>0, (t=0 is start of experiment). tt=0t=-t w A clear demonstration of time inhomogeneity: the response does not depend on t alone t Response function In e-glass the response for such a simple history, (the event at -t w ) can be described by f(t/t w ) (full aging) Is there a model that can explain time-inhomegeneity and full aging? black part simulates history, red part is experiment,.

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<t w 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! T. Grenet et. al., Eur. Phys. J. B 56, 183 (2007), and A. Amir, Y. Oreg and Y. Imry, Phys. Rev. Lett. 103, (2009) more formally, show that if the path at t>0 backtracks exactly (microscopically) the path during 0>t>- t w, one obtains f(t/t w )=ln(1+t w /t). ~ln(1+t w /t) fitted to data at small t/t w t/t w  G/G (%) ~ ln(1+t w /t)

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

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

4. Logarithmic relaxation theory  exp(-wt) W= exp(-wt) exp(-w m t)

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: slow fast e e e spirit of final state

slow (2-electron) decay

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

fast (1-electron) decay

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. e e slow t fast e ghost of initial state fast looks like a vertical dropoff on lnt

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

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 { s; w m -1 } p(w|E)<<1 ? As E decreases collective transitions become more dominant. Is this relaxation still logarithmic? Comments on 4.