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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
ln(t)
ln(t)
EDC
ln(t)
Experimental background
+
’
’’ Osheroff and coworkers (1993-2007)
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 long-range 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)
zzzz SSUSSH 21122211ˆ
|| 11 E
Two interacting TLS
Correction to the density of states (single TLS excitations)
|| 21211zSUE
No interaction: With interaction:
Correction to TLS D. o. S.
001)()( PngEnEP
TU
TTU
T
UETU
T
TU
TTU
T
UETU
T
SUEEP z
22/
cosh2
exp2
2/cosh
2exp
|)2/|(2
2/cosh
2exp
22/
cosh2
exp2
2/cosh
2exp
|)2/|(2
2/cosh
2exp
|)|()(
12121212
1211212
12121212
1211212
21211
No interaction:
With interaction:
TE
TUE
TUE
T
EPEPddgEP
2cosh
2cosh
2cosh
ln2
))()(()(
2
1212
012120
U12>>T EUEUgEP ||||2)( 121220
))()(()( 012120 EPEPddgEP
Change in D. o. S.:
Explanation of D. o. S. reduction (Efros, Shklovskii, 1975)
E1=E E2=2
E12=E+2-U12
0<2<U12-E
Instability PI~g0(U12-E), P ~-PPI
Total correction to the D. o. S.
TU
tot EUEUgEP||,2
121220
12
||||2)(
.S-h
,),(
0z
TLS
00
xS
PP
This correction should be averaged over TLS statistics (Anderson, Halperin, Warma; Phillips, 1972)
0
Sz=1/2 Sz= -1/2
Average correction to the D. o. S.
min0
max000
0
0
))/((
302
0
||,21212
20
lnln3
4
||||2)(
min0
3/10
12
E
TE
UUPP
d
r
UdP
EUEUgEP
ETEU
a
TUtot
r
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
min0000 ln
)(ln
3
4)0,(
E
TE
EUPPtEP DC
tot
Time dependent D. o. S.
t
t
TE
EUPP
t
TE
EUPPtEP
DC
DCtot
max000
min0
0000
ln)(
ln3
2
)(ln
)(ln
3
4)0,(
At time t only slow TLS’s contributes ttA
)(
120
1
Calculation of dielectric constant(adiabatic response at low temperature)
TT
S x
2tanh
3d
)(
2tanh
,)S-(h
20
2
2
320
2
20
2
2
20
2220
2
0z
TLS
μ
F
Fμ
Fμ
)(ln
3
2tanh)(
3
max2
0
20
2
02/32
02
20
00
0
0
2 maxmax
tT
P
TPP
dd
01.0~lnln9
2tanhlnln
9
2
2max00
20
20
2/
02/32
02
20
0
0
/
0
max
00
20
T
E
t
tUP
P
T
Edd
t
t
UPP
DC
DC
EE DCDC
Non-equilibrium dielectric constant
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/(gh)1/d (d – problem dimension)
• Conductivity can be approximated as
0
40
0
ln)(
~)(
~
10~,/exp~/exp~
g
g
g
g
T
arT
hhh
hh
Non-equilibrium D. o. S. and conductivity
.ln22
||||2),(
max2
1
0
)/(
20
212120
max2/1
t
teePd
r
ed
P
UUPg
tg
h
t
t
e
a
hhh
h
r
.ln2
2 max2
1
0
t
t
T
eeP h
h
Comparison with experiment
• Change in conductivity (logarithmic relaxation rate)
meV3~ln~ ,D1~ ,105~
10~2
2ln
ln
042
0
22
1
0
TP
T
eeP
td
d
h
h
h
Estimate agrees with experiment !
Width of the cusp VG
meV3~~)( hGF VE
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 non-equilibrium 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)
T
EP
T
P
TE
dd
PE
DC
LZ
DC
DC
LZ
2tanhln
3ln
3
2tanh
)(3)(
max2
0max2
0
20
2
2/320
2
20
0
0
0
20
max
0
max
Theory predicts a huge non-equilibrium effect comparable to the equilibrium one
Time dependence
2/1
20max
20
20
2
20
20
2
20
2
20
20
2
2/320
2
20
0
0
0
20
)1(
1
2tanhln
3ln
3
exp2
)(tanhexp1
2tanh
)(3),(
max
0
max
BTtT
ETP
T
P
BtT
EBt
T
E
dd
PtE
DC
LZ
DC
DC
DC
LZ
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.
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