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Interpretation of the T2-law in heat capacities of helium submonolayers
Volume 38A, number 6
INTERPRETATION
PHYSICS LETTERS
OF THE T2-LAW IN H E A T HELIUM SUBMONOLAYERS A.
BLANDIN
and
G.
13 March 1972
CAPACITIES
OF
TOULOUSE
Laboratoire de Physique des Solides*, Facult@ des Sciences, 91 - Orsay, France
Received 28 January 1972
An interpretaion of the heat capacity measurements for low coverage He submonolayers on Ar-plated Cu is given in terms of the statistica! theory of energy levels.
In a r e c e n t p a p e r , S t e w a r t and Dash [1] have p r e s e n t e d t h e i r r e s u l t on heat capacity m e a s u r e m e a t s of s u b m o n o l a y e r h e l i u m 3 and h e l i u m 4 a d s o r b e d on a r g o n - p l a t e d c o p p e r , t o g e t h e r with a c r i t i c a l r e v i e w of the v a r i o u s explanations adv a n c e d d u r i n g the l a s t decade. We wish to s u g g e s t h e r e a new i n t e r p r e t a t i o n of t h e i r r e s u l t s , which s e e m s able to r e s o l v e s o m e d i f f i c u l t i e s r e l a t e d to the l o w - c o v e r a g e , low t e m p e r a t u r e T2-1aw b e h a v i o u r . F i r s t , we v i e w the s u b s t r a t e s u r f a c e in t h e i r e x p e r i m e n t s as a v e r y d i s o r d e r e d s y s t e m . As a m a t t e r of fact, the c o p p e r is not clean, the sponge is made of c o m p r e s s e d and s i n t e r e d f l a k e s , the a r g o n p l at i n g does not c o r r e s p o n d to a s t r i c t m o n o l a y e r c o v e r a g e (1.2 B E T). T h e s u b s t r a t e is t h e r e f o r e supposed to p r o v i d e along its s u r f a c e a r a n d o m p ote n t ia l f o r a h e l i u m atom. T h i s pot en t i al c o n s i s t s of a c o l l e c t i o n of l o c a l ized s i t e s with f a i r l y r a n d o m p o s i t i o n s and potential depths. A h e l i u m atom m a y tunnel f r o m one l o c a l i z e d l e v e l to another. U n d e r such r a n d o m conditions, and with an e n e r g y s c a l e A ~ 30OK f o r the l o c a l i z e d s i n g l e p a r t i c l e e x c i t a t i o n s p e c t r u m [1], the a v e r a g e d e n s it y of l e v e l s may be e x p e c t e d to v a r y s m o o t h l y on an e n e r g y r a n g e of one d e g r e e Kelvin. The second m a i n point of o u r i n t e r p r e t a t i o n c o m e s f r o m the s t a t i s t i c a l t h e o r y of e n e r g y l e v e l s [2]. Due to l e v e l r e p u l s i o n , the s i n g l e p a r t i c l e l e v e l - c o r r e l a t i o n function, f o r the p r o b a b i l i t y of finding a l e v e l at an e n e r g y AE above the e n e r g y E o of a g i v e n l e v e l , is dropping to z e r o f o r low AE. F o r a given constant l e v e l de nsit y p = 1/5, the n o r m a l i z e d l e v e l c o r r e l a t i o n function is, roughly speaking, constant and
* Laboratoire associ~ au C.N.R.S.
equal to unity f o r AE >> 6 but exhibits a l i n e a r b e h a v i o u r f o r AE << 5 [2] **. In t h e i r study of the l i n e a r s p e c i f i c heat of g l a s s e s , A n d e r s o n , H a l p e r i n and V a r m a [3] have set a s i d e the effects of l e v e l repulsion. F r o m a flat l e v e l - c o r r e l a t i o n function, a l i n e a r law f o r the s p e c i f i c heat e n s u e s naturally. T h i s is t r u e i f T > 5 . In the o t h e r r e g i m e , if T < 6 , the l i n e a r law of the l e v e l c o r r e l a t i o n function e n t a i l s a T2-1aw f o r the s p e c i f i c heat. T h e n c e , the o b s e r v e d b e h a v i o u r will be obtained if the h y p o t h e s i s T < 5 can be justified. We can o f f e r no p r o o f at this st ag e but s e v e r a l a r g u m e n t s . We think p l a u s i b l e that the d i m e n s i o n a l i t y does f a v o r the l e v e l r e p u l s i o n ef f ect s in the helium s u b m o n o l y e r s y s t e m s as c o n t r a s t e d to the t h r e e d i m e n s i o n a l g l a s s e s c o n s i d e r e d by A n d e r s o n et al. If we c o n s i d e r the l e v e l c o r r e l a t i o n function f o r two l e v e l s at a d i s t a n c e R, as a function of AE, it is going to be a m o r e o r l e s s u n i v e r s a l function with a typical e n e r g y 5(R) N Aexp(-KR). I n t e g r a tion o v e r sp ace will give m o r e weight to distant l e v e l s ff t h e r e a r e m o r e of them. Hence, o t h e r things being equal, the e f f e c t i v e 5 will be s m a l l e r in 3 d i m e n s i o n s than in 2 d i m e n s i o n s . Th e s m a l l m a s s of helium is also a c o n t r i b uting f a c t o r b e c a u s e it tends to i n c r e a s e the e n e r g y s c a l e A and to d i m i n i s h K in the tunneling exponent, t h e r e b y i n c r e a s i n g the e f f e c t i v e 6. F i n a l l y , an a p o s t e r i o r i a r g u m e n t s t e m s f r o m c o m p a r i s o n with the d i f f i c u l t i e s linked to p r e v i o u s i n t e r p r e t a t i o n s . In the t w o - p a t c h model, the T 2law is due to t w o - d i m e n t i o a a l solid phonons in the l o w - c o v e r a g e r e g i m e ( c o v e r a g e < 0.2 monol a y e r ) as well as in the h i g h - c o v e r a g e r e g i m e . ** This last statement is true for the orthogona[ ensemble, which is the relevant one for the system considered here. 383
Volume 38A, number 6
PHYSICS LETTERS
This requires a puzzlingly large He-He interaction and l e a d s to a s u r p r i s i n g r a t i o of Debye t e m p e r a t u r e s f o r the two h y p o t h e s i z e d solid phases. In our i n t e r p r e t a t i o n , the l o w - c o v e r a g e p h a s e is a p h a s e of non i n t e r a c t i n g He a t o m s d i s p e r s e d on a fie ld of r a n d o m s i t e s : s i n g l e - p a r t i c l e l e v e l r e p u l s i o n e f f e c t s b e c o m e m a n i f e s t at the l o w e s t t e m p e r a t u r e s . Th e l o w - c o v e r a g e heat c a p a c i t y is e x p e c t e d to depend s o m e w h a t on the s u b s t r a t e d i s o r d e r and on the length of t i m e o v e r which e x p e r i m e n t s a r e done [3]*. T h i s may be the s o u r c e of d i s c r e p a n c i e s b e t w e e n d i f f e r e n t s t u d i e s [4, I]. As f o r the h i g h - c o v e r a g e , it may be a solid ~)hase, with c o l l e c t i v e phonons contributing the T ' - l a w [1]. E x p e r i m e n t a l r e s u l t s have been r e c e n t l y obtained f o r h e l i u m s u b m o n o l a y e r s a d s o r b e d on g r a p h i t e [5] which is a much " c l e a n e r " s u b s t r a t e . * The time constant is going to determine the radius of the circle of accessible levels to one helium atom. In this picture, one sees clearly the analogy with the small particle effect mentioned in ref. [2]. ~****
384
13 March 1972
In this c a s e , H e - H e i n t e r a c t i o n (or s t a t i s t i c s ) e f f e c t s dominate. T h i s s e e m s to open up the p o s s i b i l i t y of o b s e r v i n g all i n t e r m e d i a t e r e g i m e s b e t w e e n d i s o r d e r - d o m i n a t e d and i n t e r a c t i o n d o m i n a t e d s u b m o n o l a y e r s on suitable s u b s t r a t e s . It would be i n t e r e s t i n g to i n v e s t i g a t e if the a r g u e d tendency t o w a r d s s t r o n g e r l e v e l - r e p u l s i o n e f f e c t s in two d i m e n s i o n s does show up in o t h er disordered surface phases.
References [1] G.A. Stewart and J.G. Dash, Phys. Rev. A2 (1970) 918. [2] C. E. Porter, Statistical theory of spectra (Academic Press, 1965); M. L. Mehta, Random matrices (Academic Press, 1967); L. P. Gorkov and G. M. Aliashberg, Soviet Physics JETP 21 (1965) 940. [3] P.W. Anderson, B. I. Halperin and C. M. Varma, preprint. [4] W. D. Mc Cormick, D. L. Goodstein and J. G. Dash, Phys. Rev. 168 (i968) 249. A linear law for low coverages was reported. [5] M. Bretz and J.G. Dash, Phys. Rev. Letters 26 (1971) 963.
INTERPRETATION
PHYSICS LETTERS
OF THE T2-LAW IN H E A T HELIUM SUBMONOLAYERS A.
BLANDIN
and
G.
13 March 1972
CAPACITIES
OF
TOULOUSE
Laboratoire de Physique des Solides*, Facult@ des Sciences, 91 - Orsay, France
Received 28 January 1972
An interpretaion of the heat capacity measurements for low coverage He submonolayers on Ar-plated Cu is given in terms of the statistica! theory of energy levels.
In a r e c e n t p a p e r , S t e w a r t and Dash [1] have p r e s e n t e d t h e i r r e s u l t on heat capacity m e a s u r e m e a t s of s u b m o n o l a y e r h e l i u m 3 and h e l i u m 4 a d s o r b e d on a r g o n - p l a t e d c o p p e r , t o g e t h e r with a c r i t i c a l r e v i e w of the v a r i o u s explanations adv a n c e d d u r i n g the l a s t decade. We wish to s u g g e s t h e r e a new i n t e r p r e t a t i o n of t h e i r r e s u l t s , which s e e m s able to r e s o l v e s o m e d i f f i c u l t i e s r e l a t e d to the l o w - c o v e r a g e , low t e m p e r a t u r e T2-1aw b e h a v i o u r . F i r s t , we v i e w the s u b s t r a t e s u r f a c e in t h e i r e x p e r i m e n t s as a v e r y d i s o r d e r e d s y s t e m . As a m a t t e r of fact, the c o p p e r is not clean, the sponge is made of c o m p r e s s e d and s i n t e r e d f l a k e s , the a r g o n p l at i n g does not c o r r e s p o n d to a s t r i c t m o n o l a y e r c o v e r a g e (1.2 B E T). T h e s u b s t r a t e is t h e r e f o r e supposed to p r o v i d e along its s u r f a c e a r a n d o m p ote n t ia l f o r a h e l i u m atom. T h i s pot en t i al c o n s i s t s of a c o l l e c t i o n of l o c a l ized s i t e s with f a i r l y r a n d o m p o s i t i o n s and potential depths. A h e l i u m atom m a y tunnel f r o m one l o c a l i z e d l e v e l to another. U n d e r such r a n d o m conditions, and with an e n e r g y s c a l e A ~ 30OK f o r the l o c a l i z e d s i n g l e p a r t i c l e e x c i t a t i o n s p e c t r u m [1], the a v e r a g e d e n s it y of l e v e l s may be e x p e c t e d to v a r y s m o o t h l y on an e n e r g y r a n g e of one d e g r e e Kelvin. The second m a i n point of o u r i n t e r p r e t a t i o n c o m e s f r o m the s t a t i s t i c a l t h e o r y of e n e r g y l e v e l s [2]. Due to l e v e l r e p u l s i o n , the s i n g l e p a r t i c l e l e v e l - c o r r e l a t i o n function, f o r the p r o b a b i l i t y of finding a l e v e l at an e n e r g y AE above the e n e r g y E o of a g i v e n l e v e l , is dropping to z e r o f o r low AE. F o r a given constant l e v e l de nsit y p = 1/5, the n o r m a l i z e d l e v e l c o r r e l a t i o n function is, roughly speaking, constant and
* Laboratoire associ~ au C.N.R.S.
equal to unity f o r AE >> 6 but exhibits a l i n e a r b e h a v i o u r f o r AE << 5 [2] **. In t h e i r study of the l i n e a r s p e c i f i c heat of g l a s s e s , A n d e r s o n , H a l p e r i n and V a r m a [3] have set a s i d e the effects of l e v e l repulsion. F r o m a flat l e v e l - c o r r e l a t i o n function, a l i n e a r law f o r the s p e c i f i c heat e n s u e s naturally. T h i s is t r u e i f T > 5 . In the o t h e r r e g i m e , if T < 6 , the l i n e a r law of the l e v e l c o r r e l a t i o n function e n t a i l s a T2-1aw f o r the s p e c i f i c heat. T h e n c e , the o b s e r v e d b e h a v i o u r will be obtained if the h y p o t h e s i s T < 5 can be justified. We can o f f e r no p r o o f at this st ag e but s e v e r a l a r g u m e n t s . We think p l a u s i b l e that the d i m e n s i o n a l i t y does f a v o r the l e v e l r e p u l s i o n ef f ect s in the helium s u b m o n o l y e r s y s t e m s as c o n t r a s t e d to the t h r e e d i m e n s i o n a l g l a s s e s c o n s i d e r e d by A n d e r s o n et al. If we c o n s i d e r the l e v e l c o r r e l a t i o n function f o r two l e v e l s at a d i s t a n c e R, as a function of AE, it is going to be a m o r e o r l e s s u n i v e r s a l function with a typical e n e r g y 5(R) N Aexp(-KR). I n t e g r a tion o v e r sp ace will give m o r e weight to distant l e v e l s ff t h e r e a r e m o r e of them. Hence, o t h e r things being equal, the e f f e c t i v e 5 will be s m a l l e r in 3 d i m e n s i o n s than in 2 d i m e n s i o n s . Th e s m a l l m a s s of helium is also a c o n t r i b uting f a c t o r b e c a u s e it tends to i n c r e a s e the e n e r g y s c a l e A and to d i m i n i s h K in the tunneling exponent, t h e r e b y i n c r e a s i n g the e f f e c t i v e 6. F i n a l l y , an a p o s t e r i o r i a r g u m e n t s t e m s f r o m c o m p a r i s o n with the d i f f i c u l t i e s linked to p r e v i o u s i n t e r p r e t a t i o n s . In the t w o - p a t c h model, the T 2law is due to t w o - d i m e n t i o a a l solid phonons in the l o w - c o v e r a g e r e g i m e ( c o v e r a g e < 0.2 monol a y e r ) as well as in the h i g h - c o v e r a g e r e g i m e . ** This last statement is true for the orthogona[ ensemble, which is the relevant one for the system considered here. 383
Volume 38A, number 6
PHYSICS LETTERS
This requires a puzzlingly large He-He interaction and l e a d s to a s u r p r i s i n g r a t i o of Debye t e m p e r a t u r e s f o r the two h y p o t h e s i z e d solid phases. In our i n t e r p r e t a t i o n , the l o w - c o v e r a g e p h a s e is a p h a s e of non i n t e r a c t i n g He a t o m s d i s p e r s e d on a fie ld of r a n d o m s i t e s : s i n g l e - p a r t i c l e l e v e l r e p u l s i o n e f f e c t s b e c o m e m a n i f e s t at the l o w e s t t e m p e r a t u r e s . Th e l o w - c o v e r a g e heat c a p a c i t y is e x p e c t e d to depend s o m e w h a t on the s u b s t r a t e d i s o r d e r and on the length of t i m e o v e r which e x p e r i m e n t s a r e done [3]*. T h i s may be the s o u r c e of d i s c r e p a n c i e s b e t w e e n d i f f e r e n t s t u d i e s [4, I]. As f o r the h i g h - c o v e r a g e , it may be a solid ~)hase, with c o l l e c t i v e phonons contributing the T ' - l a w [1]. E x p e r i m e n t a l r e s u l t s have been r e c e n t l y obtained f o r h e l i u m s u b m o n o l a y e r s a d s o r b e d on g r a p h i t e [5] which is a much " c l e a n e r " s u b s t r a t e . * The time constant is going to determine the radius of the circle of accessible levels to one helium atom. In this picture, one sees clearly the analogy with the small particle effect mentioned in ref. [2]. ~****
384
13 March 1972
In this c a s e , H e - H e i n t e r a c t i o n (or s t a t i s t i c s ) e f f e c t s dominate. T h i s s e e m s to open up the p o s s i b i l i t y of o b s e r v i n g all i n t e r m e d i a t e r e g i m e s b e t w e e n d i s o r d e r - d o m i n a t e d and i n t e r a c t i o n d o m i n a t e d s u b m o n o l a y e r s on suitable s u b s t r a t e s . It would be i n t e r e s t i n g to i n v e s t i g a t e if the a r g u e d tendency t o w a r d s s t r o n g e r l e v e l - r e p u l s i o n e f f e c t s in two d i m e n s i o n s does show up in o t h er disordered surface phases.
References [1] G.A. Stewart and J.G. Dash, Phys. Rev. A2 (1970) 918. [2] C. E. Porter, Statistical theory of spectra (Academic Press, 1965); M. L. Mehta, Random matrices (Academic Press, 1967); L. P. Gorkov and G. M. Aliashberg, Soviet Physics JETP 21 (1965) 940. [3] P.W. Anderson, B. I. Halperin and C. M. Varma, preprint. [4] W. D. Mc Cormick, D. L. Goodstein and J. G. Dash, Phys. Rev. 168 (i968) 249. A linear law for low coverages was reported. [5] M. Bretz and J.G. Dash, Phys. Rev. Letters 26 (1971) 963.