SOME PUZZLES ABOUT LOGARITHMIC RELAXATION AND A FEW POSSIBLE RESOLUTIONS

M. PollakDept. 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

Helpfuldiscussions:AmirFrydmanOrtuñoOvadyahuThanks!

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(-Eh/kT-r/)

Eh 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

]!/)()1()ln([)()exp(1

~)( nntwtwwtdwtwt

tE nm

nm

twm

There must exist some cutoff minimal rate wm 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(-wmt) (smallest w)

exp(-wt)w=10-12

10-4

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.

-10 -5 0 5 10

6.12

6.16

6.20

6.24

6.28

6.32

Vg

2

Vg

1

G (a

rb. u

nits

)

Vg(V)

t=4.5h

t=7.5h

t=2.5h

t=1.1h

t=0.5h

t=0.15h

t=0

Vg1

Vg2

time log t

dip

am

plit

ud

e o

f G

log

Vg2Vg

1

Protocol (Ovadyahu) observed conductance(Vg,t) evolution of dips

Same as relaxation experiment

logwm-10

log(wm-1)=2log, wm

-1= 2G(t)G(t)-G0

many hours

Go

is m

ea

sure

d a

t th

is t

ime

log t0

traces staggered for clarity

2. What can tell us?

If should relate to relaxation to equilibrium then G0 must be the equilibrium G.

-10 -5 0 5 10

6.12

6.16

6.20

6.24

6.28

6.32

Vg

2

Vg

1

G (a

rb. u

nits

)

Vg(V)

t=4.5h

t=7.5h

t=2.5h

t=1.1h

t=0.5h

t=0.15h

t=0

Vg1

Vg2

time log t

dip

am

plit

ud

e o

f G

log

Vg2Vg

1

Protocol (Ovadyahu) observed conductance(Vg,t) evolution of dips

Same as relaxation experiment

logwm-1?

0

log(wm-1)=2log, wm

-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)-G0

t01secmany hours

Grenet and Delahaye, PRB 85, 235114 (2012)

Arguments that equlibrium G0 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(62r/)=10-12 exp(48)=109sec=O(10years)

Experimental results of dependence on concentration: below

100

101

102

103

104

105

106

107

108

109

1010

n (cm-3)

(se

c.)

0 1020 2.1020

Qua

si e

rgo

dic

extrapolation from ergodic regime

wm

(se

c)

1020

1018

1016

1014

1012

1010

108

106

104

102

1day

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, 235114 (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 likelyAlmost 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 experimentsSome 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

Comments on 2.

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 HUGE.

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 likelyAlmost 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 experimentsSome 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)

100 101 102 103 104 105

tw (sec.)

302274013037

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 tw

and measure response at t>0, (t=0 is start of experiment).

tt=0t=-tw

A clear demonstration of time inhomogeneity: the response does not depend on t alone

t

Re

spo

nse

fu

nct

ion

In e-glass the response for such a simple history, (the event at -tw) can

be described by f(t/tw) (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<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!

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 that if the path at t>0 backtracks exactly (microscopically) the path during 0>t>-tw , one obtains f(t/tw)=ln(1+tw/t).

~ln(1+tw/t) fitted to data at small t/tw

0

2

4

6

8

10 -4 10 -3 10 -2 10 -1 100 101 102 103

t/tw

G/G

(%

)

~ ln(1+tw/t)

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 -a at t > tw

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 {-tw,0} i.e. replacing wm by1/tw

1 10 100 10001 10 100 10000

2

tW

(sec.)

42 132 336 1000

G/G

(%)

t/tW

1 10 100 10000.0

0.5

1.0

1.5

1 10 100 10000

G/G

(%

)

t/tW

Microscopic reversibility vs. Poisson distribution of n(z)~ln(1+tw/t) ~ t-a

a=0.7 a=0.55

1 10 100 10000.0

0.2

0.4

0.6

0.8

1 10 100 10000.0

0.2

0.4

0.6

0.8

A. Vaknin et. al., PRL 84,3402 (2000)

tW (s)

35 130 740 3000

t/tw

G/G

(%

)

a=0.8

A.Vaknin et. al.,PRB 65, 2002 V. Orlyanchik & Z. Ovadyahu, PRL, 92, 066801 (2004)

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?wmtw

-1 seems physically more justifiable and in keeping with the relaxation theory.

4. Logarithmic relaxation theory

exp(-wt)W=10-12

10-4

exp(-wt)

exp(-wmt)

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

eslow

tfast

e ghost of initial state

fast looks like a vertical dropoff on lnt

t

relaxation with w=1/tw>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 {10-12s; wm-1}

p(w|E)<<1? As E decreases collective transitions become more dominant.

Is this relaxation still logarithmic?

Comments on 4.