Transcript Some Problems with Logarithmic Relaxation Theory, and a Few

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!
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, 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
There must exist some cutoff minimal rate wm below which n(w) drops off very rapidly.

1
exp( wt )d ( wt )  [  ln( wm t )   (1) n (wm t ) n / nn!]
wt
wmt
E (t ) ~  
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)
10-4
 exp(-wt)
w=10-12
(sum of future relaxations)
t=1/w
M. Pollak, M. Ortuño and A. Frydman, The Electron Glass, Cambridge University Press, 2013
Measuring the rate of decay.
the two-dip experiment on an MOS structure: gate-insulator-eglass.
Protocol (Ovadyahu)
observed conductance(Vg,t)
Vg
G (arb. units)
t=0
Vg
6.24
1
dip amplitude of G
6.28
Vg2
Vg1
evolution of dips
6.32
2
t=0.15h
t=0.5h
6.20
t=1.1h
t=2.5h
t=4.5h
6.16
Vg1
Vg2
Same as relaxation experiment
t=7.5h
6.12
Go is measured at this time
many hours
time
-10
-5
0
5
10
Vg(V)
0
log t0
log
logwm-1
log t
traces staggered for clarity
G(t)G(t)-G0
log(wm-1)=2log,
wm-1= 2
2. What can  tell us?
Protocol (Ovadyahu)
observed conductance(Vg,t)
Vg
G (arb. units)
t=0
Vg
6.24
1
dip amplitude of G
6.28
Vg2
Vg1
evolution of dips
6.32
2
t=0.15h
t=0.5h
6.20
t=1.1h
t=2.5h
t=4.5h
6.16
Vg1
Vg2
Same as relaxation experiment
t=7.5h
6.12
many hours
time
-10
-5
0
5
Vg(V)
G(t)G(t)-G0
10
0
t01sec
log
logwm-1?
log(wm-1)=2log,
log t
wm-1= 2
If  should relate to relaxation to equilibrium then G0 must be the equilibrium G.
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)
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
extrapolation
from ergodic regime
10
1020
10
1018
10
8
1016
10
7
1014
10
6
10
5
10
4
10
3
10
2
104
10
1
102
10
0
Quasi ergodic
 (sec.)
after PRL, 81, 669 (1998)
9
1012
1010
108
time from
106
cool-down?
0
20
10 -3
n (cm )
20
2.10
age of Terra
wm (sec)
10
1year
T. Grenet and J. Delahaye, Phys. Rev. B 85, 235114 (2012)
1day
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 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, 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
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 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, 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.
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).
t=-tw
t=0
t
black part simulates history, red part is experiment, .
Response function
tw (sec.)
3022
740
130
37
0
10
1
10
2
10
3
10
4
10
5
10
t
A clear demonstration of time inhomogeneity:
8
the response does not depend on t alone
7
G/G (%)
6
5
In e-glass the response for such a simple history, (the event at -tw) can
be described by f(t/tw) (full aging)
4
3
2
Is there a model that can explain time-inhomegeneity and full aging?
1
0
-4
10
-3
10
-2
10
-1
10
t/tw
0
10
1
10
2
10
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
if the path at t>0 backtracks exactly (microscopically) the path during 0>t>tw , one obtains f(t/tw)=ln(1+tw/t).
formally, show that
8
~ ln(1+tw/t)
~ln(1+tw/t) fitted to data at small
t/tw
G/G (%)
6
4
2
0
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
2
10
3
t/t w
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!
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 -a at t > tw
How does it compare with the other theory ?
Microscopic reversibility vs.
~ln(1+tw/t)
Poisson distribution of n(z)
~ t-a
1.5
2
0.8
tW (s)
A.Vaknin et. al.,PRB 65, 2002
35
130
740
3000
42
132
336
1000
A. Vaknin et. al., PRL 84,3402 (2000)
0.6
a=0.7
G/G(%)
0.4
a=0.55
1.0
G/G (%)
a=0.8
G/G (%)
V. Orlyanchik & Z. Ovadyahu,
PRL, 92, 066801 (2004)
tW (sec.)
0.5
0.2
0.0
0
0
0.0
1
10
100
t/tw
1000
1
10
100
t/tW
1000
1
10
100
1000
t/t
W
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.
4. Logarithmic relaxation theory
exp(-wmt)
exp(-wt)
10-4
 exp(-wt)
W=10-12
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
slow
fast
EXAMPLE:
e
e
e
spirit of final state
slow (2-electron) decay
After a relaxation to a new state, further relaxation to next state can be faster (larger w)
e
e
fast (1-electron) decay
e
ghost of initial state
e
e
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.
t
slow
fast
fast looks like a vertical dropoff on lnt
E
relaxation with w=1/t
w>1/t relaxations
from state at E
with probability p(w|E)
exp(-wt), w=1/t
t
Is this
relaxation still logarithmic?
log t
Comments on 4.
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.