Short Notes K107

phys. stat. sol. (b) 154, K107 (1989) Subject classification: 62.65 and 65.70; 63.20

Solid State Physics Research Centre, Presidency College, Calcutta' ) (a) and Physics Department, Jadavpur University, Calcutta2 ) (b) On the Nature of the Distribution Function for Two-Level Systems in Amorphous Solids

BY SUBRATA MUKHERJEE (a) , S. SENGUPTA (a) , and A.N. BASU (b)

I t is well known that experiments on some amorphous solids such as vitreous silica, borosilicate glass, etc. indicate I1 I anomalous properties in specific heat, thermal conductivity, dispersion, and absorption of acoustic waves at temperatures below about 1 K . A broad understanding of these anomalies could be achieved by assuming a distribution of two-level systems (TLS) in the solid as originally

suggested by Anderson et al. 121 and Phillips I31 and developed further by Jackle et al. 14, 51 and Hunklinger and Arnold 161. The basis idea is that within the solid local clusters exist with closely lying aouble minima in the configuration space. The two fundamental parameters associated with these are the energy difference between the two minima 2 A and the overlap A. between the two states. The Hamiltonian matrix for the system can be written as

which on diagonalisation becomes

where

E = W

and A is the unitary eigenvector matrix. When the solid is strained both A and A.

are changed and assuming a linear response we write

For simplicity we consider the effect of only one strain component e. The Hamiltonian under strain is obtained from (1) by writing A' , A: in place of A ,

A o , Performing the unitary transformation with the matrix A we get the perturbed Hamiltonian

l ) Calcutta 700013, India. 2 , Calcutta 700032, India.

K108

De 2Me

2Me -De H = H o t H 1 , H = l

where

physica status solid (b) 154

9

In the application of this theory, so far discussed by different authors, a distribution of TLS energies E is assumed but D and M are taken as more or less constant quantities. Recently measurement of the Griineisen constant I' from thermal expansion results / 7 / gave a value of the order of 100, while according to the above phenomenological theory r sD/kT X 10 ( D s 1 eV) at T = 1 K . Measurement of the pressure dependence of acoustic relaxation absorption by Bartell and Hunklinger / 8 / also leads to trouble. Under a static pressure of 1 kbar, De is eV so that all TLS will be so much raised in energy that no absorption of acoustic pulses is expected to occur. Experiments, however, show no such effect. To get over these difficulties, it has been suggested that there may be a distribution of D values over the TLS ranging from positive to negative values.

In this note we take up the question of distribution of both D and E values in some details. The first thing we want to assert is that we must introduce a distribution of the two basic parameters of a TLS, A and Ao. The distribution in E and D will follow as a consequence.

We can imagine the distribution of TLS over a two-dimensional space spanned by the Cartesian co-ordinates A and Ao. The probability that a two-level system will have its parameters between A , A. and A + dA, P(A, Ao)dA dAo,

where P(A, A o ) is the distribution function. Going over to the polar co-

ordinates (E, 8 ) where A = EcosB, ho = Esin8 we can wri te the probability as P(E,B)EdEdB. We assume A , Ao>O and the ranges of variation for E and 0 are 0 to Emax and 0 to n / 2 , respectively. Since specific heat and other data indicate a more or less constant density of TLS over E , we take the distribution function to be of the form P(E , 6) = (A/E)f(B). Integrating over all angles we get, for the probability of a TLS to lie between the energy range E and E t dE,

4

A. t dAo is

T / 2 P(E)dE = A / f(8)d8 dE .

0

In absence of any direct knowledge about the nature of the function f ( O ) , we expand it in terms of Legendre polynomials. Thus

f(B) = C an Pn(cose) (n = 0, 1, 2, ...) , ( 3 ) n

where the an are the coefficients to be determined from the boundary conditions of

Short Notes K109

the problem. In our case the function should vanish for vanishing A

(large potential barrier or configurational distance) and should be maximum for A

= 0 (symmetric TLS). Satisfying these conditions we can develop f (e ) by truncating the series at n = 2 without any free parameter. The function is

0

(4a) 2 f(8) sin 8

with n = 3 we have a single free parameter in f (e) , (4b) 2 3 f ( e ) = ( a + 3)sin e + Scos 8-2 ,

where a = 3ao/5a3. The maximization condition requires a + 3>0. The distribution function can only be tested by application to different

experimental results which we shall presently do using the distribution function in (4a). Determining A from normalisation, we finally write the probability function

" 4 sin'e P(E,B) = -- E n nEmax

(5)

The number density

n(E, e ) = NP(E,

where N total number

function of TLS can be written in the form

8 ) 9 ( 6 )

of TLS per unit volume and no = NIEmax is the constant density of TLS per unit energy interval.

Application We write D = a cose + b sine and N = (1/2)(a sine - b cos8)

where we put a = dA/de, b = dAo/de and treat both as constants. Note that D can

also be expressed as dElde. The contribution to the Gruneisen parameter' from TLS of type ( E , 8 ) is given

by / 9 / (symbols are the same as in the reference),

(as iav) , n(E, 8)EdEdO

C dT =

and

The total contribution from all TLS is X T I 2

I 2 m d x sech xdx r = - - 1 KT (a cos8 + b sine)f(e)de 0 0

where x = E/ZkT and xm = Emax/2kT. Assuming xm>>l , and we get

the rirst integral is In 2

8 physica (b)

K l l O physica status solid (b) 154

Tr = -20 K [?‘I = - 5.6 (a + 2b) 3 a K for f ( 8 ) given by (4a).

Acoustic relaxation and resonance absorption experiments at low temperature can determine the ratio of no<DIM1> and no<M1> / 5 / . The reference shows large experimental uncertainty and the values are 0.94 eV and 0.55 eV for two different experiments. With the help of ( 4 ) we determine this ratio as

2 2 2

2 2

C D ~ 2 2 M1 > ;? TI a 2 2 b + 32 71 (a4 - 6a 2 2 b + b 4 ) + g 1 ab(a2 - b2) ~

- ” a2 + 2 b2 - 1 ab <MF - 16 16 (10)

for the first distribution function. Solving (9 ) and (10) we get the values of a and b , which are

a = 23.028 eV and b = j1.514 eV . (11)

Interpretation of the phonon echo experiment Using our postulate (5) we can now proceed to explain the phonon echo experiment / l o / . With varying pulse areas for the input pulses Graebner and Golding have measured the pulse area of the echo pulse. The echo intensity due to the TLS lying in the range E to E + dE and 0 to 0 + de is given by /11/

2 2 4 Iecho a (dn) sin 4 sin ( 4 1 2 )

where 4 denotes the pulse area of

eOT @ = - (a sine - b cose) . 2 h

9 (12)

two equal input pulses and is given by

Considering the pulse time of the echo as a constant, the pulse area of the echo depends on

2 n(E, e)sin$sin ( g l 2 )

for these specified TLS. The echo pulse area of the whole system is thus

sin [; (a sine - b cos8) (a sine - b cos8) f(e)de , (14) 1 n/2 c L f

0

where

x = e ~ / h .

Calculation Fig. 1 shows the result of our calculation of the phonon echo area with (14) compared to the experimental curve / l o / . For postulate (4a) the curve

Short Notes K l l l

Fig. 1. Phonon echo pulse area. The continuous line refers to the experiment / l o / . The dash-dotted curve is calculated from (14) with f(8) = sin 8

2

agrees completely with experiment upto the maxima of the experimental curve but shows fast oscillations after that. To see if the second distribution function given in (4b) reduces the strong oscillations in the echo pulse area, we used arbitrary values of the parameter a and found no marked improvement even with large variation in a. The absence of oscillations in the experimental curve seems to indicate some

inadequacy in the present theory because, as is clear from (14), due to the presence of the sine functions, oscillations with respect to x cannot be avoided.

The broad agreement indicates that the distribution function f ( 8 ) removes the strong disagreement between thermal and acoustic experiments. The shape of the curve and the position of the peak are strongly sensitive to the values of a and b. Thus accurate measurement of the ratio <D M >/<M > is necessary for a correct prediction of the distribution function.

Discussion The anomaly regarding the acoustic absorption at extremely high pressure ( x 1 kbar) is not really a problem of the distribution of TLS energies. If we write the energy gap of a TLS under a static volume strain en in the form

2 2 2

E + dE = /(A + aeo) 2 + ( A o + beo) 2' ,

then it is implied that A>>aeo and Ao>>beo, as otherwise the structure of the original TLS will be completely destroyed. Thus the real anomaly is the fact that even when the pressure is of the order of 1 kbar and eo= TLS are not a t all destroyed but as shown by the resonance absorption experiments 181, maintain their identity wit9 marginal modification. This perhaps indicates that the response of TLS to dynamic strain is different from response to static strain so that the response coefficients a , b for static strain are much less than a , b for dynamic strain.

Conclusion In this note our aim was to show that the TLS model can successfully explain all the thermal and acoustic experiments including the thermal expansion results with a single set of TLS parameters. The form of the distribution function f(8) cannot be uniquely determined due to lack of accurate experimental results. The note emphasizes the effect of the parameter dAo/de in different experiments. We also conclude that a new theory is needed to explain the effect of static external strain.

K112 physica status solid (b) 154

References /1/ W .A. PHILLIPS (Ed. ) , Amorphous Solids: Low-Temperature Properties,

/ 2 / P.W. ANDERSON, B . I . HALPERIN, and C.M. VERMA, Phil. Mag. 25, 1

/ 3 / W.A. PHILLIPS, J. low-Temp. Phys. 1, 351 (1972). 141 J. JACKLE, Z. Phys. 257, 212 (1972). / 5 / J. JACKLE, L. PICHE, W. ARNOLD, and S. HUNKLINGER, J. non-

Spr inge r -Ver l ag ,1981.

(1972).

crystall. Solids g, 365 (1976).

Ed. W.P. MASON and R.N. THURSTON, Academic Press, New York 1976. / 6 / S. HUNKLINGER and W. ARNOLD, in: Physical Acoust ics , Vol. 12 ,

/ 7 / D.A. ACKERMAN et al., Phys. Rev . B 29, 966 (1984). / 8 / U. BARTELL and S. HUNKLINGER, J. Physique 43, C9-489 (1982). / 9 / W.A. PHILLIPS, J. low-Temp. Phys. 11, 757 (1973).

/ l o / J . E . GRAEBNER and B. GOLDING, Phys. Rev . B l9, 964 (1979). /11/ I .D. ABELLA, N.A. KURNIT, and S .R . HARTMANN, Phys. Rev . 141,

391 (1966).

(Received January 23, 1989)