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.