Transcript Theory of Slow Non-Equilibrium Relaxation in Amorphous

Theory of Slow Non-Equilibrium
Relaxation in Amorphous Solids
at Low Temperature
Alexander Burin
Tulane, Chemistry
Outline
• Experimental background and theory goals
• Pseudo-gap in the density of states (D.o.S.)
• Break of equilibrium and induced changes
in D.o.S.
• Non-equilibrium dielectric constant and
hopping conductivity within the TLS model
• Conclusions
• Other mechanisms of non-equilibrium
dynamics
Experimental background
EDC
+
ln(t)

’
Osheroff and coworkers
(1993-2007)
’’
ln(t)

ln(t)
Ovadyahu and coworkers
(1990-2007), Grenet and
coworkers
(2000-2007),
Popovich and coworkers
(2005-2007)
Goals
• Interpret experimental observations in terms of
the non-equilibrium raise of the density of states
of relevant excitations (TLS or conducting
electrons) with its subsequent slow relaxation
backwards
• The changes in the density of states are
associated with the “Coulomb gap” effects
induced by TLS – TLS or TLS – electron longrange interactions
Non-equilibrium dynamics
-
External force raises density of
states for relevant excitations
Slow relaxation lowers D. o. S.
back to equilibrium
Case of study: TLS in glasses
(Burin, 1995)
Two interacting TLS
Hˆ  1S1z   2 S 2z  U12 S1z S 2z
Correction to the density of states (single TLS
excitations)
No interaction:
E1 | 1 |
With interaction:
E1 | 1  U S |
z
12 2
Correction to TLS D. o. S.
No interaction:
P( E)  n   ( E  1 )  ng0  P0
P1 ( E )   ( E  | 1  U12 S 2z |) 
With

interaction:
 
   U12 / 2 
exp 2  cosh 1
 ( E  | 1  U12 / 2 |)
2T
 2T 


 2 
 1  U12 / 2 
 2 
 1  U12 / 2 
exp  cosh
  exp 
 cosh

2T
2T
 2T 


 2T 


 2 
 1  U12 / 2 
exp 
 cosh
 ( E  | 1  U12 / 2 |)
2T
 2T 



 
   U12 / 2 
 2 
 1  U12 / 2 
exp 2  cosh 1

exp

cosh





2T
2T
 2T 


 2T 


Change in D. o. S.:




 P( E )  g02  d1  d 2 ( P1 ( E)  P0 ( E ))




 P ( E )  g 02  d1  d 2 ( P1 ( E )  P0 ( E )) 

 E  U12 
 E  U 12  
 cosh 2T  cosh 2T  




 2T ln 
2 E 


cosh 



 2T 
U12>>T 
 P(E)  2g02 | U12 | E | U12 | E
Explanation of D. o. S. reduction
(Efros, Shklovskii, 1975)
E2=2
E1=E
E12=E+2-U12
0<2<U12-E
Instability
PI~g0(U12-E), P ~-PPI
Total correction to the D. o. S.
 Ptot ( E )  2 g
2
0

2 ,|U12 | T
| U12 |  E | U12 |  E 
This correction should be averaged over TLS statistics
(Anderson, Halperin, Warma; Phillips, 1972)
P
P (,  0 ) 
,
0
hˆ TLS  -Sz   0 S x .
Sz=1/2
Sz= -1/2
0

Average correction to the D. o. S.
 Ptot ( E )  2 g 02
 P
2
0
(U 0 /( E T ))1 / 3

a

2 ,|U12 | T
| U12 |  E | U12 |  E  
U0
d 0
U max
4
E
dr 3 

P0 P0U 0 ln
ln
r  0 min  0
3
E  T  0 min
E
Since P0U0~10-3 we have P << P.
Change in D. o. S due to external
DC field application
Energy shift E = -FDC/, ~3D, FDC~10MV/cm, E~7K >> T
Only TLS with E<E can be removed out of equilibrium
EDC
4
E
 Ptot ( E, t  0)  
P0 P0U 0 ln
ln
3
( E  T )  0 min
Time dependent D. o. S.
At time t only slow TLS’s contributes
1
1  2  t
A 0 (t )
EDC
 0 (t )
4
 Ptot ( E , t  0)  
P0 P0U 0 ln
ln
3
( E  T )  0 min
EDC
t max
2

P0 P0U 0 ln
ln
3
( E  T )
t
Calculation of dielectric constant
(adiabatic response at low temperature)
hˆ TLS  -(  Fμ)Sz   0 S x ,
 2  2
0
   tanh
 2T

 2 

3
 max
2
  2 (  Fμ) 2  2 μ 2

0
0



dF 2
3 2

  20

 max
 d 
0
0
d 0
20
( P0  P)
0
2  20
P0   2    max 

ln
   (t )
3
 T 


3/ 2

3
2
 2  2
0
tanh
 2T

 2  2
0
tanh
 2T









Non-equilibrium dielectric constant
2P0   2 
  
P0U 0
9
t max
 ln
t

E DC /  E DC / 
 d 
0
P0   2 
9
0
d 0

 0 2  20

2
0

3/ 2
2
2




EDC
0
ln
tanh
 2T


t max 2 EDC
P0U 0 ln
ln
~ 0.01
t
T




Non-equilibrium conductivity
(Burin, Kozub, Galperin, Vinokur, 2007)
EF
Variable range hopping
• Defined by charges with energy h>T (h~Ta,
a=3/4, Mott; a=1/2, Efros, Shklovskii)
• Hopping to the distances rh~1/(gh)1/d (d –
problem dimension)
• Conductivity can be approximated as

~ exp  h / T  ~ exp rh / a ,  0 ~ 104 
0
  h g ( h ) g ( h )   0 
~
~
ln 

T
g
g
 
Non-equilibrium D. o. S. and
conductivity
g ( h , t )
g
 2 P0  | U12 |  h | U12 |  h 
2
( e /  h )
1/ 2
P0

2

a
e
dr 2
r
t max

t
1
2
P0 e  e   tmax 

 ln
 2
.

   h   t 
d
1
2
P0 e  e   h  tmax 



 2
ln
.

  h  2T  t 
Comparison with experiment
• Change in conductivity (logarithmic relaxation
rate)
1
2
P0 e  e   h
d ln 


 2
~ 102
d ln t
   h  2T
P0  2

0 
~ 5 10 ,  ~ 1D,  h ~ T ln  ~ 3meV
 
4
Estimate agrees with experiment !
Width of the cusp VG
EF (VG ) ~  h ~ 3meV
Estimate agrees with the experiment!
(Vaknin, Ovadyahu, Pollak, 2002)
Suggestion
• Investigate glassy properties in related
materials, i. e. temperature dependence
of sound velocity and/or sound attenuation
and dielectric constant temperature
dependence at T<1K.
Conclusions
• TLS model can be used to interpret nonequilibrium relaxation in glasses and
doped semiconductors
• The
non-equilibrium
relaxation
is
associated with the evolution of the
density of states affected by the long –
range interaction (Coulomb or dipolar gap)
Acknowledgement
• Support by Louisiana Board of Regents,
contract no. LEQSF (2005-08)-RD-A-29)
• Tulane
University
Research
and
Enhancement Funds
• To organizers of this extraordinary
workshop for inviting me
Interaction unrelated non-equilibrium
dielectric constant
(Yu and coworkers, 1994; Burin 1995)
P0   2 
 ( EDC ) 
3
 max
 max
 d 
0
 0 LZ
d 0
20
 0 (  EDC ) 2  20


3/ 2
 2  2
0
tanh
 2T





P0   2    max  P0   2    max 
 EDC 



ln
ln
tanh



3
3
 T 
 2T 
  LZ 
Theory predicts a huge non-equilibrium effect
comparable to the equilibrium one
Time dependence
P0   2 
 ( EDC , t ) 
3

 2  2
0
  tanh
 2T



 max
 max
 d 
0
 0 LZ
d 0
20
 0 (  EDC ) 2  20


3/ 2
2

 (  E ) 2  2




DC
0
0
1  exp  Bt



tanh

2
2 





2T
0 




P0   2    max  P0   2   T

ln
ln

3
3
 T 
  LZ
2





0
 exp  Bt


2
2 

  0 




1
 E 
 tanh DC 
1/ 2
 2T  (1  BTt)

Power law relaxation is associated with
interaction stimulated dynamics (Burin, Kagan,
1994) only so one can study it. Better materials
are those which have no nuclear quadrupole, i. e.
mylar.