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THE LOCAL PHONON DENSITY OF STATES AND
ITS INFLUENCE ON SUPERCONDUCTING
PROPERTIES
G. Arnold, M. Menon
To cite this version:
JOURNAL DE PHYSIQUE
CoZi;oque C6, suppZdrnent au n o 12, Tome 42, d6eembre 1981 page C6-377
THE LOCAL PHONON DENSITY OF STATES A N D ITS INFLUENCE ON SUPERCONDUCTING
PROPERTIES
G.B. Arnold and M. Menon
Univ. of Notre-Dame, Dept. of Physics, Nofire-Dame, IN 46556,
U.
S. A.A b s t r a c t . We have c a l c u l a t e d t h e l o c a l phonon d e n s i t y of s t a t e s and l o c a l e l e c t r o n phonon coupling parameters f o r a t h i n metal f i l m backed by a semi- i n f i n i t e metal s u b s t r a t e , using t h e Montroll-Potts model. We f i n d t h a t only a small enhancement of t h e l o c a l e l e c t r o n phonon coupling i n t h e f i l m c a n be produced by t h e coupling of t h e f i l m l a t t i c e t o t h e s u b s t r a t e . Q l l a l i t a t i v e l y , t h e l o c a l d e n s i t y of s t a t e s i n t h e t h i n f i l m i s l i k e t h a t of t h e i s o l a t e d film, except f o r some smearing of s h a r p s t r u c t u r e , and a g e n e r a l downward s h i f t i n frequency of t h e g r o s s f e a t u r e s .
I n t h i s paper, we w i l l c o n s i d e r t h e e f f e c t of a semi-inf i n i t e f i l m (S) on t h e l o c a l phonon d e n s i t y of s t a t e s i n a t h i n (21 l a y e r s ) f i l m ( T I , and determine whether
S can thereby i n f l u e n c e t h e electron-phonon i n t e r a c t i o n
(BPI)
i n T. Only t h e case f o r which S and T a r e good m e t a l s w i l l be considered.The e l e c t r o r p h o n o n coupling s t r e n g t h i s measured by t h e McMillan parameter1
X = 2 f ° J a 2 ( o ) ~ ( o ) l o (1)
where a'(@) i s a convolution of t h e e l e c t r o n band d e n s i t y of s t a t e s and t h e square of EPI matrix elements, and P(o) i s t h e phonon d e n s i t y of s t a t e s . One can s i m i l a r l y d e f i n e a l o c a l s t r e n g t h parameter f o r t h e mth l a y e r of a metal f i l m
I n t h i s paper we s h a l l focus on t h e l o c a l phonon d e n s i t y of s t a t e s f o r t h e m t h l a y e r of metal f i l m T. FT(o,m). I n o r d e r t o determine t h e e f f e c t of changes i n FT(o,m) on t h e l o c a l EPI coupling, we s h a l l approximate cr2(o.m) by a frequency independent parameter a:
.
Because of t h e l o c a l i t y of t h e screened EPI and Coulomb i n t e r a c t i o n , one can a l s o d e f i n e a l o c a l p a i r p o t e n t i a l i n t h e t h i n f i l m . a s t h e r a t i o ~ ( o , m f = ~ ( o , m ) / Z y ( o , m ) where +(o,m) i s t h e p a i r i n g s e l f energy and ZT(o,m) i s t h e r e n o r m a l i z a t i o n f u n c t i o n i n T, each depending on t h e l o c a l a:~~(w,m) . a
For s i m p l i c i t y , we have adopted t h e Montroll-Potts modela f o r t h e phonons. We have f u r t h e r assumed t h a t both S and T l a t t i c e s a r e simple cubic w i t h equal l a t t i c e c o n s t a n t s , and t h a t t h e l a t t i c e s a r e a l i g n e d . with i n t e r f a c e p e r p e n d i c u l a r t o t h e
C6-378 JOURNAL DE PHYSIQUE
[I001 d i r e c t i o n . The method of Maradudin and Wallis,' a s a p p l i e d by Dobryznski and M i l l s s t h e n allows t h e exact phonon Green's f u n c t i o n i n T t o be c a l c u l a t e d . The e f f e c t i v e s p r i n g c o n s t a n t of t h e i n t e r f a c e plane i s K, and t h e s p r i n g c o n s t a n t s f o r S and T a r e K and KT. We d e f i n e an e f f e c t i v e coupling parameter y = K ~ / ( K & ) . The r a t i o of t h e maximum phonon frequency i n T t o t h a t i n S i s denoted by R.
The r e s u l t s f o r t h e t r a n s m i s s i o n c o e f f i c i e n t .
i t
i2,
f o r phonons normally i n c i d e n t from T i s shown i n f i g u r e (1). One can show a n a l y t i c a l l y t h a tit i 2
i s a maximum f o r any o when y = l . Phonons of a l l f r e q u e n c i e s i n T can be t r a n s m i t t e d t o S only i f t h e maximum phonon frequency i n T i s l e s s t h a n t h a t i n S, i.e., R<1.f a c e between two m a t e r i a l s with various values o f coupling, y.
The dashed curves a r e f o r R=3, t h e s o l i d curves a r e f o r R=.30. The frequency W T . (21(T/MT)
' I 2 ,
MT being t h e atom mass f o r T.0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
W / O ,
I n f i g u r e ( 2 ) we compare t h e
total
d e n s i t y of s t a t e s i n b u l k T m a t e r i a l t o t h a t i n a n i s o l a t e d ( y 4 ) T metal film. I n g e n e r a l , q T ( w ) i s s h i f t e d down i n frequency f o r t h e b u l k ( y = l , el) c a s e , b u t , even so, oT(1/2cr2)=.640 f o r t h e i s o l a t e d f i l m , .643 f o r t h e b u l k metal, a n e g l i g i b l e d i f f e r e n c e .1.28,
Fig. 2. T o t a l d e n s i t y of s t a t e s f o r bulk T m a t e r i a l (dashed l i n e ) compared with same f o r 21 l a y e r f i l m of T m a t e r i a l ( s o l i d l i n e ) .
w / w ,
i n t e r f a c e l i e s between t h e l a y e r s m=-1 and m=O.
Fig. 3. Local density of states for R = . 3 (solid lines) conpared to same for R=3 (dashed lines)
.
For optimal coupling of S and T, we f i n d l o c a l EPI parameters AmzwT(~a/2ai)=.649 ( ~ 0 ) . .644(m=1), . 6 4 4 ( ~ 2 ) . For weak ST coupling Am=.623(m=0), .639(m=1). .642(m=2). The d i f f e r e n c e i s g r e a t e s t a t t h e ST i n t e r f a c e , r a p i d l y diminishing a s one mwes away i n t o T.
These r e s u l t s show t h a t t h e l o c a l p a i r p o t e n t i a l i n T w i l l have e n e r m dependence which i s q u i t e d i f f e r e n t from t h e bulk T dependence, b u t very much l i k e t h a t p r e d i c t e d f o r t h e i s o l a t e d T metal f i l m except f o r downward s h i f t s i n
frequency and some smearing of peaks. I n a d d i t i o n , t h e l o c a l energy gap v a l u e (determined by A,,, and
g )
w i l l d i f f e r n e g l i g i b l y frcm i t s v a l u e f o r t h e i s o l a t e d f i l m . Thus, even under optimal c o n d i t i o n s , t h e coupling between t h e T l a t t i c e and t h e S l a t t i c e p r o v i d e s l i t t l e enhancement of l o c a l superconducting p r o p e r t i e s i n a T metal f i l m of t h i s t h i c k n e s s (21 l a y e r s ) . We have a l s o v e r i f i e d t h i s c o n c l u s i o n f o r a T metal f i l m c o n s i s t i n g of only four atomic l a y e r s , where we have found t h a tA m=.648(m=0), . 6 4 3 ( ~ 1 ) , .639(m=2), . 6 2 2 ( ~ 3 ) . T h i s work was supported by NSP g r a n t DMR80-19739. References
1. W.L. McMillan, Phys. Rev.
