- Email: [email protected]
Ultra-slow processes in disordered insulators
Joumal of Non-Crystalline Solids 77 & 78 (1985) 131-134 North-Holland, Amsterdam
131
ULTRA-SLOW PROCESSES IN DISORDERED INSULATORS Michael POLLAKand Allen G. HUNT* Dept. of Physics, University of California, Riverside CA 92521, USA We consider some consequences of very slow transition rates in Anderson localized disordered systems. Primary focus is on the glassy state of the electronic system. The problem is considered without, and with Coulomb interactions. I t is found that the entropy changes smoothly through the glass temperature, while the poarizability changes sharply. Some possible applications of the electron glass are discussed. I . INTRODUCTION The existence of localized states in solids which lack the translational symmetry of crystals leads to the hopping transport mechanism, f i r s t clearly proposed by MottI and by Conwell2. Such transport differs qualitatively from the more t r a d i t i o n a l l y considered band transport by Bloch electrons in a number of aspects3. Here we mainly consider the consequences of two. Transition rates can be very small even when they are downwardin energy, and they can have a very wide distribution. The nearly linear frequency dependence of the conductiv i t y at small frequencies ~, and the so-called dispersive transport are consequences of these aspects. The p o s s i b i l i t y to shift the entire distribution of relaxation times by orders of magnitude by varying the temperature is another important aspect for the effects discussed below. At low temperature T the rates may become so slow that the electronic system becomes non-ergodic, i . e . i t may not be able to reach thermal equilibrium during experimental times. This state of the electron system has been termed the electron glass. Such sluggish behavior is usually not associated with electrons. Though the electron glass can exist also in the absence of interactions, i t is much more l i k e l y to occur when Coulomb interactions are important. 2. CONSEQUENCES OF SLOWTRANSITION RATES. Dispersive transport 4 and ac conductivity5 ~(m) are discussed extensively in the l i t e r a t u r e . Here we commentonly on their direct relation to slow rates. Very frequently6o is observed to be nearly proportional to ~ down to very small w.Transition rates comparable to m then must s t i l l be important in transport. *Present address: Fachbereich Fysik, Phillips Universit~t, Marburg, F.R.Germany 0022-3093/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
M. Pollak, A.G. Hunt / Ultra-slow processes
132
Otherwise Reo (~) would have to be an even function, as required by the analytic properties of ~ at m~O. The time domain experiments (e.g. photoluminescence) are more complicated because "far from equilibrium " conditions prevail. However,the frequently observed power-lawtime dependence at long times t again signifies that processes with rates comparableto t - l must be important. We now turn to the electron glass. There have been two approaches to this problem. One employs an order parameter7, as in much of the work on spin glass. In essence such a method tests whether the system can get to equilibrium at t -~. The other method8 tests whether equilibrium properties are observable after a reasonable experimental time. Such a time can be defined sharply enough by using the logarithmic time scale y = In(wot) , (I) where w° is some elementary rate, usually a phonon frequency, Wo~lOl3 sec" I . The practical range of experimental times Ye is narrow; for lO sec y=18, for l day y:24. Observability of equilibrium in an experiment is identified with the abil i t y to diffuse over macroscopic distances after a time Ye" As w i l l be seen,this criterion
implies that a f i n i t e fraction of the system has reached equilibrium.
We now turn to transition rates, at f i r s t in the absence of interactions. I f wij is the transition rate from an i n i t i a l state i to a final state j , then the actual rate of transitions from i to j is uij: fi(l-fj)wij , (2) where fk is the occupation probability of state k; wij clearly depends on the mechanism for transitions.The important point is that w i j , and often f i ' are exponential functions of random variables. We define a random variable x i j , u.. (3) 13 : WoeXp('xij ), or x i j = In(wo/Ulj) . Equations (1) and (3) show that y and x are naturally compatible quantities. The logarithmic scale permits the approximation that for a time y all transitions with xijY. Macroscopic diffusion at Ye exists i f the bonds defined by xij
fi(O) is given by the equilibrium at T+AT. When AT~T, we ar~
"near equilibrium", since fi(O) differs l i t t l e from the equlibrium value of fi at T. We can thus use the l a t t e r in eq.(2).Assuming that at low T transitions occur by single-phonon assisted hopping, x i j : 2 r i j / a + Eij/kT ,
(4)
M. Pollak, A.G. Hunt / Ultra-slow processes
where r i j
is the distance between states i , j ,
133
a is t h e i r localization radius,
and E.. is an appropriately defined9 energy relating to the pair i , j . The perlj colation condition is given in terms of an xc such that when bonds are defined by xij
depends on the density 0%
states. For a constant density of states ~ near theCFermi level,xc=2.3(~a3kT)- - . Macroscopic diffusion during an experiments is possible when Ye>Xc. The onset of this condition happens at the glass temperature T , hence Tg =(2.5/Ye)4/(~a3k) : 2x~o-n/(~a3k)
(5)
Below Tg electrons can diffuse only microscopic distances, while above Tg macroscopic diffusion is possible.ln fact, for y>xc the system can be devided in two parts; the states on the percolation cluster, which constitutes a f i n i t e fraction of the system, is in thermal equilibrium, while the remainder is not. Usually the observable properties arise mainly from the percolation cluster,when i t exists. We can then say that above T the system exhibits equilibrium properg t i e s , while below T i t does not. g We now wish to examine whether some property, s p e c i f i c a l l y the entropy per site s, and the e l e c t r i c a l p o l a r i z a b i l i t y x change sharply at T .For simplicity g we take a r a t i o K=I/2 of electrons to the total number of states N t . l f G is the total number of configurations available to the system at y, then s(y)=klnG/kT, where k is the Boltzman constant. The bonds Y>Xij form clusters such that there are CN clusters of N sites. Any such cluster with n electrons contributes a factor (nN)zN~/[n~(N-n)t] to G. At y>xc one cluster has an i n f i n i t e number of states N~.For K=I/2 its contribution to InG is N In2. The behavior near Tg is given by the behavior when Ye is near Xc.The increase As in s from YcZXc to y near but above Yc is
As : N~In21Nt where the f i r s t
~ (CNINt ) Kn (I-K) N-n (nN) In(nN) - A's, (6) N,n term is the entropy o f the i n f i n i t e c l u s t e r , the second term is
the entropy at Yc o f those clusters which are merged into the i n f i n i t e
cluster
at y, i.eo ENCN:N~,and the t h i r d term is the change in entropy from Ye to y o f those clusters which are s t i l l
f i n i t e at y. Clearly, i f there is a sharp change
in s at Yc' i t must come from the f i r s t cancel e x a c t l y . But S t i r l i n g ' s
two terms. Using S t i r l i n g ' s
formula they
formula is not accurate f o r small N. I f we denote
z2-N(nN)In(nN)zln2 ND(N), then D(N) decreases n
r a p i d l y to zero as N increases,
ks = ~ D(N)/N t Apparently AS comes mainly from those sites on the percolation cluster which formed s_mal__llclusters at y_. Since the population o f small clusters varies smoothly through c r i t i c a l percolation I 0 , AS varies smoothly through Yc' and therefore at Tg . 11
(7)
M. Pollak, A. G. Hunt / Ultra-slow processes
134
To obtain X(Ye) we can calculate the polarization a time Ye after the application nf a small f i e l d ~, while keeping the temperature constant. This gives x = Z~eAfir i.F/F.FV~ . (8) The f i r s t sum is over clusters, the second over states in a cluster, r i is the position of a state measured from the "center of gravity" of i t s cluster, V the volume, and Afi=e)ci.~/kT the change induced by ~ after Ye" The change in x at Tg must be sharp, because some X]i diverge in the i n f i n i t e cluster. From the cluster s t a t i s t i c s lO and from the form of eq. (8), x=(yc-y) -l 66 We can only summarize here b r i e f l y our work on the system with Coulomb interactions. At low densities the main effect of interactions is to scramble the levels of the system states so that "low lying states are related by transfers of many electrons, and many-electron hopping becomes important. Improving on a previous percolation theory for this process, we evaluated T and the behavior of s g and x at T . The improvement in the percolation theory raised T by about an org g der of magnitude. The entropy again varies smoothly, and × abruptly, with an exponent -2.4, at T . Relevant experiments on the electron glass do not yet exist, g but observations on spin glasses seem to show that there some properties change abruptly, others vary smoothly at Tg. We do not know whether the other approach to the electron glass can lead to similar conclusions, but we believe that the behavior of the glass at any t depends on processes with rates l / t ,
not I / = .
The existence of many states, some possibly of high energy, from which decay to equilibrium is extremely slow at low T, may have application to energy storage and memory. Studying the behavior with i n i t i a l conditions "far from equilibrium" w i l l be important for this purpose. The basis for the properties we discussed is lack of translational symmetry. Someof the results thus may apply to biological systems, large aperiodic heterostructures, and other devices. REFERENCES l ) N.F. Mott, Can. J. Phys. 34 (1956) 1356. 2) E.M. Conwell, Phys. Rev. I03 (1956) 51. 3) M. Pollak, Proc. Int. Conf. Semicond. Phys., Exeter 1962, p. 86. 4) For a recent review see J. M. Marshall, Rep. Prog. Phys. 46 (1983) 1235 . 5) For a recent review see A.R. Long, Adv. Phys, 31 (1982) 553 . 6) A.K. Jonscher, Nature 253 (1975) 717. 7) M. Grbnewaldet. a l . , J. Phys. C 15 (1982) Lll53; J. H. Davies, P.A. Lee, and T.M. Rice, Phys. Rev. Lett 49 (1983) 758 . 8) M. Pollak, Phil. Mag. 50 (1984) 265. g) M. Pollak, J. Non-cryst. Solids I I (1972) I . IO) D. Stauffer, Phys. Rep. 54 (Ig7g) I . l l ) for more detail see M. Pollak and A. G. Hunt, Phil. Mag., in press.
131
ULTRA-SLOW PROCESSES IN DISORDERED INSULATORS Michael POLLAKand Allen G. HUNT* Dept. of Physics, University of California, Riverside CA 92521, USA We consider some consequences of very slow transition rates in Anderson localized disordered systems. Primary focus is on the glassy state of the electronic system. The problem is considered without, and with Coulomb interactions. I t is found that the entropy changes smoothly through the glass temperature, while the poarizability changes sharply. Some possible applications of the electron glass are discussed. I . INTRODUCTION The existence of localized states in solids which lack the translational symmetry of crystals leads to the hopping transport mechanism, f i r s t clearly proposed by MottI and by Conwell2. Such transport differs qualitatively from the more t r a d i t i o n a l l y considered band transport by Bloch electrons in a number of aspects3. Here we mainly consider the consequences of two. Transition rates can be very small even when they are downwardin energy, and they can have a very wide distribution. The nearly linear frequency dependence of the conductiv i t y at small frequencies ~, and the so-called dispersive transport are consequences of these aspects. The p o s s i b i l i t y to shift the entire distribution of relaxation times by orders of magnitude by varying the temperature is another important aspect for the effects discussed below. At low temperature T the rates may become so slow that the electronic system becomes non-ergodic, i . e . i t may not be able to reach thermal equilibrium during experimental times. This state of the electron system has been termed the electron glass. Such sluggish behavior is usually not associated with electrons. Though the electron glass can exist also in the absence of interactions, i t is much more l i k e l y to occur when Coulomb interactions are important. 2. CONSEQUENCES OF SLOWTRANSITION RATES. Dispersive transport 4 and ac conductivity5 ~(m) are discussed extensively in the l i t e r a t u r e . Here we commentonly on their direct relation to slow rates. Very frequently6o is observed to be nearly proportional to ~ down to very small w.Transition rates comparable to m then must s t i l l be important in transport. *Present address: Fachbereich Fysik, Phillips Universit~t, Marburg, F.R.Germany 0022-3093/85/$03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
M. Pollak, A.G. Hunt / Ultra-slow processes
132
Otherwise Reo (~) would have to be an even function, as required by the analytic properties of ~ at m~O. The time domain experiments (e.g. photoluminescence) are more complicated because "far from equilibrium " conditions prevail. However,the frequently observed power-lawtime dependence at long times t again signifies that processes with rates comparableto t - l must be important. We now turn to the electron glass. There have been two approaches to this problem. One employs an order parameter7, as in much of the work on spin glass. In essence such a method tests whether the system can get to equilibrium at t -~. The other method8 tests whether equilibrium properties are observable after a reasonable experimental time. Such a time can be defined sharply enough by using the logarithmic time scale y = In(wot) , (I) where w° is some elementary rate, usually a phonon frequency, Wo~lOl3 sec" I . The practical range of experimental times Ye is narrow; for lO sec y=18, for l day y:24. Observability of equilibrium in an experiment is identified with the abil i t y to diffuse over macroscopic distances after a time Ye" As w i l l be seen,this criterion
implies that a f i n i t e fraction of the system has reached equilibrium.
We now turn to transition rates, at f i r s t in the absence of interactions. I f wij is the transition rate from an i n i t i a l state i to a final state j , then the actual rate of transitions from i to j is uij: fi(l-fj)wij , (2) where fk is the occupation probability of state k; wij clearly depends on the mechanism for transitions.The important point is that w i j , and often f i ' are exponential functions of random variables. We define a random variable x i j , u.. (3) 13 : WoeXp('xij ), or x i j = In(wo/Ulj) . Equations (1) and (3) show that y and x are naturally compatible quantities. The logarithmic scale permits the approximation that for a time y all transitions with xij
fi(O) is given by the equilibrium at T+AT. When AT~T, we ar~
"near equilibrium", since fi(O) differs l i t t l e from the equlibrium value of fi at T. We can thus use the l a t t e r in eq.(2).Assuming that at low T transitions occur by single-phonon assisted hopping, x i j : 2 r i j / a + Eij/kT ,
(4)
M. Pollak, A.G. Hunt / Ultra-slow processes
where r i j
is the distance between states i , j ,
133
a is t h e i r localization radius,
and E.. is an appropriately defined9 energy relating to the pair i , j . The perlj colation condition is given in terms of an xc such that when bonds are defined by xij
depends on the density 0%
states. For a constant density of states ~ near theCFermi level,xc=2.3(~a3kT)- - . Macroscopic diffusion during an experiments is possible when Ye>Xc. The onset of this condition happens at the glass temperature T , hence Tg =(2.5/Ye)4/(~a3k) : 2x~o-n/(~a3k)
(5)
Below Tg electrons can diffuse only microscopic distances, while above Tg macroscopic diffusion is possible.ln fact, for y>xc the system can be devided in two parts; the states on the percolation cluster, which constitutes a f i n i t e fraction of the system, is in thermal equilibrium, while the remainder is not. Usually the observable properties arise mainly from the percolation cluster,when i t exists. We can then say that above T the system exhibits equilibrium properg t i e s , while below T i t does not. g We now wish to examine whether some property, s p e c i f i c a l l y the entropy per site s, and the e l e c t r i c a l p o l a r i z a b i l i t y x change sharply at T .For simplicity g we take a r a t i o K=I/2 of electrons to the total number of states N t . l f G is the total number of configurations available to the system at y, then s(y)=klnG/kT, where k is the Boltzman constant. The bonds Y>Xij form clusters such that there are CN clusters of N sites. Any such cluster with n electrons contributes a factor (nN)zN~/[n~(N-n)t] to G. At y>xc one cluster has an i n f i n i t e number of states N~.For K=I/2 its contribution to InG is N In2. The behavior near Tg is given by the behavior when Ye is near Xc.The increase As in s from YcZXc to y near but above Yc is
As : N~In21Nt where the f i r s t
~ (CNINt ) Kn (I-K) N-n (nN) In(nN) - A's, (6) N,n term is the entropy o f the i n f i n i t e c l u s t e r , the second term is
the entropy at Yc o f those clusters which are merged into the i n f i n i t e
cluster
at y, i.eo ENCN:N~,and the t h i r d term is the change in entropy from Ye to y o f those clusters which are s t i l l
f i n i t e at y. Clearly, i f there is a sharp change
in s at Yc' i t must come from the f i r s t cancel e x a c t l y . But S t i r l i n g ' s
two terms. Using S t i r l i n g ' s
formula they
formula is not accurate f o r small N. I f we denote
z2-N(nN)In(nN)zln2 ND(N), then D(N) decreases n
r a p i d l y to zero as N increases,
ks = ~ D(N)/N t Apparently AS comes mainly from those sites on the percolation cluster which formed s_mal__llclusters at y_. Since the population o f small clusters varies smoothly through c r i t i c a l percolation I 0 , AS varies smoothly through Yc' and therefore at Tg . 11
(7)
M. Pollak, A. G. Hunt / Ultra-slow processes
134
To obtain X(Ye) we can calculate the polarization a time Ye after the application nf a small f i e l d ~, while keeping the temperature constant. This gives x = Z~eAfir i.F/F.FV~ . (8) The f i r s t sum is over clusters, the second over states in a cluster, r i is the position of a state measured from the "center of gravity" of i t s cluster, V the volume, and Afi=e)ci.~/kT the change induced by ~ after Ye" The change in x at Tg must be sharp, because some X]i diverge in the i n f i n i t e cluster. From the cluster s t a t i s t i c s lO and from the form of eq. (8), x=(yc-y) -l 66 We can only summarize here b r i e f l y our work on the system with Coulomb interactions. At low densities the main effect of interactions is to scramble the levels of the system states so that "low lying states are related by transfers of many electrons, and many-electron hopping becomes important. Improving on a previous percolation theory for this process, we evaluated T and the behavior of s g and x at T . The improvement in the percolation theory raised T by about an org g der of magnitude. The entropy again varies smoothly, and × abruptly, with an exponent -2.4, at T . Relevant experiments on the electron glass do not yet exist, g but observations on spin glasses seem to show that there some properties change abruptly, others vary smoothly at Tg. We do not know whether the other approach to the electron glass can lead to similar conclusions, but we believe that the behavior of the glass at any t depends on processes with rates l / t ,
not I / = .
The existence of many states, some possibly of high energy, from which decay to equilibrium is extremely slow at low T, may have application to energy storage and memory. Studying the behavior with i n i t i a l conditions "far from equilibrium" w i l l be important for this purpose. The basis for the properties we discussed is lack of translational symmetry. Someof the results thus may apply to biological systems, large aperiodic heterostructures, and other devices. REFERENCES l ) N.F. Mott, Can. J. Phys. 34 (1956) 1356. 2) E.M. Conwell, Phys. Rev. I03 (1956) 51. 3) M. Pollak, Proc. Int. Conf. Semicond. Phys., Exeter 1962, p. 86. 4) For a recent review see J. M. Marshall, Rep. Prog. Phys. 46 (1983) 1235 . 5) For a recent review see A.R. Long, Adv. Phys, 31 (1982) 553 . 6) A.K. Jonscher, Nature 253 (1975) 717. 7) M. Grbnewaldet. a l . , J. Phys. C 15 (1982) Lll53; J. H. Davies, P.A. Lee, and T.M. Rice, Phys. Rev. Lett 49 (1983) 758 . 8) M. Pollak, Phil. Mag. 50 (1984) 265. g) M. Pollak, J. Non-cryst. Solids I I (1972) I . IO) D. Stauffer, Phys. Rep. 54 (Ig7g) I . l l ) for more detail see M. Pollak and A. G. Hunt, Phil. Mag., in press.