SUPERCONDUCTIVITY I N ALKALI TUNGSTEN BRONZES

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SUPERCONDUCTIVITY I N ALKALI TUNGSTEN

BRONZES

S. Ruvalds, L. Kahn

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-460

SUPERCONDUCTIVITY IN ALKALI TUNGSTEN BRONZESt

J. Ruvalds and L.M. Kahn

Physios Department, University of Virginia, Charlottesville, Virginia 22901, U.S.A.

Résume.- Le comportement anormal de la température de transition supraconductrice dans quelques bronzes de tungstène alcalin peut être expliqué par le couplage des électrons induit par l'échange de plasmons acoustiques.

Abstract.- The anomalous behavior of the superconducting transition temperature in cer-tain alkali tungsten bronzes is shown to be consistent with electron pairing induced by exchange of acoustic plasmons.

The superconducting properties of certain alkali tungsten bronzes exhibit some highly unusual features which appear to violate the predictions of the BCS theory. These nonstoichiometric compounds are of the form M W 03, where M represents an Alkali atom such as Na, Kb, and hydrogen. As x increases, these compounds become metallic, and the electronic density of states is believed to increase monotoni-cally with the alkali content as indicated by ma-gnetic susceptibility measurements l\f.

On the basis of the BCS theory 111 the superconducting transition temperature T should become higher with increasing electronic density of states at the Fermi energy N(0) according to the relation

T

c

=O.70

D

ex

P

[-^Jgj-j

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Where the Debye temperature is roughly 0 ^400 K in these materials and g is a matrix element of the electron-phonon coupling. For a free electron model

1/ 3

N(0) a x , and the predicted T increases with x as shown in Figure 1. If the electron density of

states increases faster (say N(0) «x as in Refe-rence /l/),then the transition temperature of course should rise more sharply as a function of alkali content.

In sharp contrast to these expectations, the experimental data show an enhancement of T with decreasing x for Rb W 03 /3,4/ and Na W03/5/as

w x seen in Figure 1. The low values of the electronic

specific heat and magnetic susceptibility in these materials /!/ would seem to rule out spin

fluctua-tion effects, and the extrapolated density of t Supported in part by the National Science

Foun-dation Grant No DMR 77-13167

electron states is quite small, resembling the values of Na, Cu, and other non-superconducting metals.

The purpose of the present work is to correlate the anomalous T -dependence in these materials with an electron pairing mechanism achie-ved by virtual exchange of acoustic plasmons.

8 IT 1 1 1 1 T i S 7 - \ A R b „ W 03 — \ T o Na, WO,

* \t

h- \

cc 1\ Ul \

v- \

- \ I / \ t o \?o T S \ BCS < * • ' " —* ^

P 1 "

^

^ "

•^ ^ - ' o n r I I I I I I 0.1 0.2 0.3 0.4 A L K A L I CONCENTRATION X F i g . 1 : Superconducting t r a n s i t i o n temperature as a function of a l k a l i content for RbxW0 (Referen-ces / 3 , 4 / ) and NaxW03 (Reference / 5 / ) . The BCS theory using a f r e e - e l e c t r o n d e n s i t y of s t a t e s (N(o) a x 1/3 ) i s shown by the dotted l i n e . The s o l i d curve i s the r e s u l t of the present work using the Frohlich expression of Equation ( 3 ) .

This process was o r i g i n a l l y suggested by Frohlich / 6 / as a p o s s i b i l i t y i n metals with overlapping d and s - e l e c t r o n bands. Acoustic plasmon modes a r e generated by the screening of the d-plasma

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o s c i l l a t i o n s by t h e s - e l e c t r o n s , which y i e l d s a l i n e a r d i s p e r s i o n f o r t h e plasmons of t h e form

wq

'

cq ( 2.)

where t h e plasmon "sound" v e l o c i t y i s given by c=(w 2m /a 2m )1!2~s, i n terms of t h e r e s p e c t i v e

d s s d

(R=d,s) plasma f r e q u e n c i e s o Q 5 4nnR2/mQ, t h e e f f e c t i v e masses mR, and t h e Fermi v e l o c i t y v of t h e s - e l e c t r o n s . By t r e a t i n g t h e a c o u s t i c plasmons i n d i r e c t analogy with o r d i n a r y phonons, F r o h l i c h

/ 6 / d e r i v e d t h e f o l l o w i n g e x p r e s s i o n f o r t h e superconducting t r a n s i t i o n temperature

-

1 T

-

W e

- ,

F (3) w i t h

where 6 i s t h e degeneracy of t h e d-band,

ms

and md denote t h e number of e l e c t r o n s i n t h e s and d bands r e s p e c t i v e l y , and i s t h e d i e l e c t r i c c o n s t a n t , and a' = ki2/e2 (rnSmdf

' I 2 .

S i m i l a r l y , t h e correspon- d i n g plasmon cut-off temperature W can b e expressed

Where D i s t h e d e n s i t y of energy l e v e l s p e r e l e c - R

t r o n near t h e Fermi energy

The a c o u s t i c plasmon exchange mechanism f a v o r s high temperature s u p e r c o n d u c t i v i t y f o r two b a s i c p h y s i c a l reasons. F i r s t t h e coupling of e l e c t r o n s t o a c o u s t i c plasmons i s e s s e n t i a l l y a screened Coulomb i n t e r a c t i o n which may be expected t o be s t r o n g e r than t h e u s u a l electron-phonon cou- p l i n g . Secondly t h e cut-off frequency W may b e of t h e o r d e r of a plasmon frequency

(W

10 K) t h u s overwhelming t h e phonon c o n t r i b u t i o n which i n v o l - v e s a Debye energy c u t - o f f e D 400 K. On t h e o t h e r hand, t h e formation of a c o u s t i c plasmon mo- d e s d e p r i v e s c o n s i d e r a b l e s t r e n g t h from t h e bulk " o p t i c a l " plasmons, and t h e r e f o r e d e c r e a s e s t h e s c r e e n i n g of t h e Coulomb e l e c t r o n - e l e c t r o n r e p u l - s i o n . F i n a l l y i t should be emphasized t h a t t h e a c o u s t i c plasmons a r e generated only under r e s t r i c - t i v e c o n d i t i q n s , and t h e i r well-defined occurance may be r a r e .

Applying t h e F r o h l i c h f ormulas/3-5/ t o t h e d a t a f o r t h e super conducting t r a n s i t i o n temperature of a l k a l i t u n g s t e n bronzes, we o b t a i n t h e remarka-

b l e agreement f o r t h e v a i a t i o n of T . w i t h a l k a l i c o n t e n t a s shown i n F i g u r e 1 . The f i t t o experiment y i e l d s v a l u e s of t h e coupling F(x=O.1)'0.32 t o F(x=0.5)=0.17, and Wz200K. U s i n g r e a l i s t i c ~ ~ + . 4 e V ,

ms = mo,

md

= 3mo, 6 = 5 , and E = 10 y i e l d l a r g e r v a l u e s of F(x=O. I )

=

0.8 and W = l o 4 K. Therefo- r e , t h e competition from t h e screened Coulomb re- p u l s i o n cannot be n e g l e c t e d i n a q u a n t i t a t i v e a p p l i c a t i o n of t h e theory. T h i s c o n c l u s i o n i s sup- p o r t e d by numerical c a l c u l a t i o n s by P a s h i t s k i i / 7 / and o t h e r s who o b t a i n T

=

100 K from s o l u t i o n s of t h e superconducting energy gap e q u a t i o n .

There i s some p r e l i m i n a r y support f o r our band model i n t h e o p t i c a l d a t a / 8 / which r e v e a l s an a b s o r p t i o n edge a t A% 3eV f o r pure i n s u l a t i n g WOg

.

As Na _Xi s added t h e a b s o r p t t o n edge moves h i g h e r i n energy f o r

x

> 0.1, corresponding t o a t r a n s i t i o n from t h e v a l e n c e band t o t h e p a r t i a l l y f i l l e d conduction band. The s h i f t i n t h e absorp- t i o n edge i s roughly 0.3eV which i s c o n s i s t e n t with o u r s-band superimposed on a r i g i d conduction d-band whose bottom i s roughly 0.2eV above t h e s-band minimum.

I n summary, t h e a c o u s t i c plasmonmechanism t o g i v e a good account of t h e anomalous concentra- t i o n dependence of T i n a l k a l i t u n g s t e n bronzes. F u r t h e r c a l c u l a t i o n s of t h e reduced Coulomb s c r e e - ning a r e i n p r o g r e s s . F i n a l l y i t would b e i n t e r e s - t i n g t o probe t h e a l k a l i t u n g s t e n bronzes by d i - r e c t e l e c t r o n l o s s measurements; our c a l c u l a t i o n s y i e l d Landau damping of t h e a c o u s t i c modes, b u t n e v e r t h e l e s s show a s i g n i f i c a n t peak i n t h e s t r u c - t u r e f a c t o r a t f i n i t e momenta which should be o b s e r v a b l e by experiment.

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References

/I/ See for example, Greiner, J.D., Shanks, H.R., and Wallace,D.C.,

J.

Chem. Phys.

2 (1972) 772 and references cited therein.

/ 2 /

Bardeen, J., Cooper, L.N. and Schrieffer, J.R., Phys. Rev.

106 (1957) 162.

/3/ Remeika, J.P., Geballe, T.H., Matthias, B.T., Cooper, A.S., Hull, G.W.,

and Kelly, E.M., Phys. Lett.

&

(1967) 565.

141 Sweedler,

A.R.

(PhD.

Thesis, University of California, La Jolla,

California 1969).

For a recent review of T enhancement in tungsten bronzes, see I. Lefko-

witz, Ferroelectrics

3 f1977) 239.

/ 5 /

Shanks, H.R., Sol. St. Comrnun

2 (1974) 753.

/ 6 /

Frohljch,

H.,

J.

Phys. C._1 (1968) 544.

/ 7 /

Pashitskii, E.A. and Chernousenko, V.M., Soviet Physics

JET?