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) E DC ln(t) Experimental background +  ’  ’’ Osheroff and coworkers ( ) Ovadyahu and coworkers ( ), Grenet and coworkers ( ), Popovich and coworkers ( )

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) Two interacting TLS Correction to the density of states (single TLS excitations) No interaction:With interaction:

Correction to TLS D. o. S. No interaction: With interaction:

U 12 >>T  Change in D. o. S.:

Explanation of D. o. S. reduction (Efros, Shklovskii, 1975) E 1 =E E2=2E2=2 E 12 =E+  2 -U 12 0<  2 <U 12 -E  Instability P I ~g 0 (U 12 -E),  P ~-PP I

Total correction to the D. o. S. This correction should be averaged over TLS statistics (Anderson, Halperin, Warma; Phillips, 1972)

Average correction to the D. o. S. Since P 0 U 0 ~10 -3 we have  P << P.

Change in D. o. S due to external DC field application Energy shift  E = -  F DC / ,  ~3D, F DC ~10MV/cm,  E~7K >> T Only TLS with E<  E can be removed out of equilibrium

Time dependent D. o. S. At time t only slow TLS’s contributes

Calculation of dielectric constant (adiabatic response at low temperature)

Non-equilibrium dielectric constant

Non-equilibrium conductivity (Burin, Kozub, Galperin, Vinokur, 2007) EFEF

Variable range hopping Defined by charges with energy  h >T (  h ~T a, a=3/4, Mott; a=1/2, Efros, Shklovskii) Hopping to the distances r h ~1/(g  h ) 1/d (d – problem dimension) Conductivity can be approximated as

Non-equilibrium D. o. S. and conductivity

Comparison with experiment Change in conductivity (logarithmic relaxation rate) Estimate agrees with experiment !

Width of the cusp V G 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 ( )-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) Theory predicts a huge non-equilibrium effect comparable to the equilibrium one

Time dependence 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.