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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 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 ) ng 0 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 2 T 2 T 2 T 2 T Change in D. o. S.: P( E ) g 02 d1 d 2 ( P1 ( E ) P0 ( E )) P( E ) g 02 d1 d 2 ( P1 ( E ) P0 ( E )) E U12 E U12 cosh 2T cosh 2T 2T ln 2 E cosh 2T U12>>T P( E) 2 g02 | 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. P ( E ) 2 g | U | E | U | E 2 0 tot 2,|U12| T 12 12 This correction should be averaged over TLS statistics (Anderson, Halperin, Warma; Phillips, 1972) P P ( , 0 ) , 0 Sz=1/2 ĥ TLS -S 0 S . z Sz= -1/2 0 x Average correction to the D. o. S. Ptot ( E ) 2 g 02 P02 (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) ĥ TLS -( Fμ)S z 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 2P0 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/(gh)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 t max 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 P0 2 max P0 2 max ln ln 3 3 T LZ 3/ 2 2 2 0 tanh 2T E tanh DC 2T 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 0 0 LZ d 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.