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ON THE RELAXATION TIME OF CERTAIN TWO-LEVEL SYSTEMS I N AMORPHOUS
STRUCTURES
L. Ferrari, G. Russo
To cite this version:
L. Ferrari, G. Russo. ON THE RELAXATION TIME OF CERTAIN TWO-LEVEL SYSTEMS I N AMORPHOUS STRUCTURES. Journal de Physique Colloques, 1981, 42 (C4), pp.C5-1073-C5-1076.
�10.1051/jphyscol:19814235�. �jpa-00220866�
CoZZoque suppZ&ment au nO1O, Tome 4 2 , octobre 1981 page C4-1073
ON THE R E L A X A T I O N T I M E OF C E R T A I N TWO-LEVEL SYSTEMS I N AMORPHOUS STRUCTURES
L. Ferrari and G.
uss so"
I s t i t u t o d i Matematica d e l l 'Universitd - Ferrara, I t a l y
Gruppo NazionaZe d i S t r u t t u r a deZZa Materia
-
Unitd d< BoZogna-
Bologna, I t a l y"
I s t i t u t o d i Fisica del2 'Universitd-
Bologna, I t a l yGrmppo NazionaZe d i S t r u t t u r a deZZa Materia - Unitd d i BoZogna
-
BoZogna, I t a ZyAbstract.- The origin and dynamics of 2-level systems in amorphous structures have been recently interpreted by the authors in terms of D D intimate pairs
+ -
interactions. Since the "tunnelling" particles are electrons (and not atoms) and the quantum picture of the model is quite different from that first int- roduced by Anderson et Al. in 1972, the problem concerned with the relaxation time z of 2-level systems has been reviewed and z re-calculated. At given te- mperature, z is shown to be much shorter than predicted by previous theories.
It is suggested that the corresponding 2-level systems represent the "anoma- lous" (or fast) ones introduced by Black and Halperin for interpretating some time-dependent experimental data obtained at low temperature.
Introduction.- The relaxation time z is a quantjty directly related to the detailed stpucture of the 2-level systems present in many amorphous substances, hence the relevance of T for getting reliable information about the origin of 2-level systems themselves has increased more and more in recent years (1) giving rise, among other chings, to some challenging questions whose clearest example is the dependence of scjme thermodynamical quantities on the measurement time (1,2).
In a recent, still unpublished paper ("Charged dangling bonds and two-level systems in amorphous solids") the authors have developed a model for the 2-level sy- stems dynamics which strongly differs from the previous ones expecially as far as the nature of the defects corresponding to the 2-level systems is concerned.
The model deals with D D (opposite charged dangling bonds) intimate pairs,
+ -
forming electric dipoles whose allowed "motion" at low temperature, is simply a cha- rge interchange; it is suggested (3) that a certain percentage of such defects is formed at the surface of some crystalline inclusions characteristic of many amorpho- us substances ( 4 ) . Finally it has been shown that crystalline inclusions containing 2 electric dipoles may give origine to 2-level systems due to dipole-dipole intera- ctions. Behind the complicated details of the model, a clear dynamical feature is remarkable even at a first sight: the "moving" particles are no longer atoms or gro- ups of atoms, but, if any, pairs of electrons. The second remark is that the def- ect cbnnot be described in terms of a double potential well scheme, as Anderson et Al. did in 1972 (5); in the present model, in fact, 2-level systems are originated by a rather complicated coupling between charged defects, so that the calculation of z requires a different and subtler dynamical picture.
The basic formula one is concerned with is the standard one, essentially deri- ved from Fermi's Golden Rule (6):
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19814235
JOURNAL DE PHYSIQUE
where V is the perturbing potential due to the elastic strain, Il> and 12s are qu- antum states corresponding to the 2-level systems separated by the energy splitting E,and C is given in terms of the sound velocities C and C as follows:
1 t
In expression (1) the detailed structure of the 2-level systems is contained in the matrix elementcl(V12>. In the model under consideration the energy quantum E is gained (or lost) when one dipole is at rest (and, say, upward directed) whyle the second dipole "switches up" from the downward position (or vice-versa). Hence, apart from very complicated but negligeable deformation effects, the whole 2-dipole system can be approximately described in terms of the wave functions corresponding to the "switching" dipole only and hence one assumes, according to a chemical-physi- cal-picture of weak polar bonds:
where fa> and /b> are bi-electronic normalised wave functions representing D in each of the two atomic sites a and b where the D D pair is located. Ortogonality of
+ -
11s and 12s is given by the assumption:
Icalbzl 1
.
The real parameter a is related to the dipole momentum M of the electric dipole ac- cording to the formula:
Assuming that:
- R
c blrlbs =
- -
and2
expression (3) can be used for determining a as a function of M and R (the distance between the atomic sites a and b). The value of M can be determined from experiments
(7) and, for a-SiO it turnes out to be 2'
o f 4%
!
seems t o be r e l i a b l e and t h u s , from r e 1 . s ( 4 ) , ( 3 ) and ( 2 ) , one h a sA c c o r d i n g t o a s t a n d a r d p r o c e d u r e ( 6 ) , one now assumes t h a t V i s d i a g o n a l i n t h e l a s and I b > b a s i s ; t h e n a s t r a i g h t f o r w a r d c a l c u l a t i o n g i v e s ( 6 ) :
where B i s t h e d e f o r m a t i o n p o t e n t i a l w i t h a r o u g h magnitude o f 1 eV. Thus from f o r - mulas ( l ) , ( 4 ) , ( 5 ) and ( 6 ) , one f i n a l l y o b t a i n s :
E x p r e s s i o n ( 7 ) shows t h a t t h e r e l a x a t i o n t i m e deduced from t h e d i p o l a r model, f o r g i v e n T and E z K T i s comparable w i t h t h e l o w e s t v a l u e among t h e o n e s p r e d i c t e d by t h e o r i e s b a s e d on t u n n e l l i n g atoms ( 2 , 5 ) . Hence t h e c o n j e c t u r e t h a t t h e d i p o l a r B
2 - l e v e l s y s t e m s c a n c o r r e s p o n d t o t h e "anomalous" ( o r f a s t ) o n e s p r o p o s e d by Black a n d H a l p e r i n ( 8 ) f o r e x p l a i n i n g some time-dependent e x p e r i m e n t a l d a t a ( g ) , seems t o be v e r y r e a s o n a b l e . T h i s c o n j e c t u r e h a s been t e s t e d a c c o r d i n g t o d a t a deduced from r e f . ( 1 ) ( s e e i n p a r t i c u l a r pag. 4 1 ) . I t h a s b e e n found t h a t a t T=0.16 K', t h e r e l a x a t i o n t i m e f o r p r o c e s s e s i n v o l v i n g 2 - l e v e l s y s t e m s s t r o n g l y c o u p l e d w i t h t h e dominant phonon i s a b o u t 1 0 -5 s e c , i n e x c e l l e n t a g r e e m e n t w i t h t h e e x p e c t e d v a l u e f o r t h e "anomalous" l e v e l s , a s g i v e n i n r e f . (1) ( p a g . 4 1 ) . F u r t h e r m o r e i t i s n o t i - c e a b l e t h a t " a n o m a l o u s " l e v e l s have been i n t r o d u c e d w i t h o u t i n c r e a s i n g t h e v a l u e o f B; t h u s i t i s l i k e l y t h a t t h e p r e s e n t model c a n overcome t h e d i f f i c u l t i e s c o n n e c t e d w i t h a t o o s t r o n g c o u p l i n g o f "anomalous" 2 - l e v e l s y s t e m s w i t h phonons. F i n a l l y , t h e r e l a t i o n among 2 - l e v e l s y s t e m s and e l e c t r i c d i p o l e s , s u p p o r t e d by e x p e r i m e n t a l e v i - d e n c e ( 7 ) , c l e a r l y r e f l e c t s on t h e r e l a x a t i o n t i m e t h r o u g h p a r a m e t e r a 2 ; u s i n g r e l . ( 5 ) i t i s p o s s i b l e t o r e - w r i t e t h e e x p r e s s i o n f o r z - l a s f o l l o w s :
2 3 B E
c-'(M,E) = [ l - ( 2 M / e R 1 ~ 1 coth(E/2KBT)
,
c
8 n e M 4where t h e dependence o f z on M h a s b e e n made e x p l i c i t .
JOURNAL DE PHYSIQUE
References:
(1) "Amorphous Solids"(Low-Temperature properties); Editor: W.A. Phillips, SPRINGER- -VERLAG
-
Berlin (1981)(2) J. L. Black; Phys. Rev. B
12
NO6,
2740 (1978)(3) L. Ferrari and G. Russo; Lett. N. Cim.
30
NO6,
184 (1981)(4) J. Chahoud, L. Ferrari and G. Russo; J. Non-Cryst. Sol.
E&% ,
1141 (1980) (5) P.W. Anderson, B. I. Halperin and C.M. Varma; Phil. Mag.2 ,
1 (1972)( 6 ) J. Jackle; 2 . Physik 257
,
212 (1972)(7) B. Golding, M.W. Schickfus, S. Hunklinger and K. Dransfeld; Phys. Rev. Lett.
2
NO24
,
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