A phenomenological description of the fermi-surface of white tin in terms of the APW-model

PHYSICS

Volume 30A, n u m b e r 3

t h e r m a l - e x p a n s i o n v a l u e s f o r ~ In ~ / ~ In V. F o r comparison we have also given the experimental v a l u e s [1]. T h e u n c e r t a i n t i e s , w h e r e q u o t e d , a r e t h o s e a r i s i n g f r o m ~ In ~ / ~ In V. W h i l e t h e a g r e e m e n t of o u r c a l c u l a t e d r e s u l t s may be to some extent fortuitous, the consiste n c y of t h e o v e r a l l r e s u l t s i s e n c o u r a g i n g . A d e c i s i v e t e s t of t h e a b o v e r e l a t i o n h o w e v e r w i l l r e q u i r e a c c u r a t e v a l u e s of a In ~ / ~ i n V. Numerous refinements are possible in the c a l c u l a t i o n of k f r o m m i c r o s c o p i c t h e o r y [5] a n d in the expression for Tc, which is valid only in t h e w e a k - c o u p l i n g l i m i t . T h e e l a b o r a t i o n of t h e a b o v e i d e a s w i l l b e t h e s u b j e c t of a s u b s e q u e n t work.

Table 1 Elem

~b(expt)

AI Cd Zn Hg c~ Hgfl In Pb Sn Lac~ Ta TI Nb V Ti Zr Ir Re Mo U

3.3 2.8 2.2 2.1 3.9 2.2 1.9 2.4 -2.5 0.7 -0.3 0.5 -0.44 -1.9 -3.2 2.5 1.4 0.9 -27

(b1 3.4 5.7 4.7 3.3 5.1 1.0 1.9 1.2 -0.5 0.6 -1.9

q~2 1.0 1.0"2.0

± 1.5 =~2.1 ±2

±0.15 :~ 0.15 ± 0.5

-0.04 ± 0.08

6 October 1969

LETTERS

1.9 0.6 NO.6 -0.3 0.2 0.2 0.1 ~0.5 0.0 0.4 0.4 0.6 -1.4

References 1. R.I. Broughton, J. L. Oisen and C. Palmy, Prog. Low Temp. Phys. Vol. VI, to be published. 2. R. E. Hodder, Phys. Rev., to be published. 3. P. Morel and P. W. Anderson, Phys. Rev. 125 (1962) 1263. 4. P . E . Seiden, Phys. Rev. 179 (1969) 458. 5. P . E . Selden, Phys. Rev. 168 (1968) 403.

e l e m e n t s [1]. W e h a v e t h u s l i s t e d i n s e p a r a t e c o l u m n s ( s e e t a b l e 1) t h e v a l u e s (~bl) c a l c u l a t e d using the pressure-effect results for In 3//a In V a n d t h o s e (~b2) o b t a i n e d u s i n g t h e * * * * *

A PHENOMENOLOGICAL OF WHITE

TIN

DESCRIPTION IN TERMS OF

OF THE

THE FERMI-SURFACE APW-MODEL

M. A. C. D E V I L L E R S a n d A. R. D E V R O O M E N

Faculty of Science, Katholieke Universiteit, NijnTegen, The Netherlands Received 30 August 1969

It is shown e x p e r i m e n t a l l y that a muffin-tin potential, t r e a t e d r e l a t i v i s t i c a l l y , can d e s c r i b e the F e r m i s u r f a c e g e o m e t r y of a n e a r l y f r e e e l e c t r o n m e t a l to within the b e s t available e x p e r i m e n t a l a c c u r a c y as s e t by r e c e n t G a n t m a k h e r e x p e r i m e n t s .

It h a s b e e n s u g g e s t e d m a n y t i m e s , b u t n e v e r carried out, that the APW-scheme could lend itself excellently as an interpolation scheme for a Fermi-surface. We tested this on the rather complicated Fermi-surface of w h i t e t i n , w e l l known by recent radio frequency size effect m e a s u r e m e n t s of [1]. T h e i n f o r m a t i o n a b o u t t h e m u f f i n - t i n p o t e n t i a l i s c o n t a i n e d in t h e d i m e n s i o n l e s s ot I a n d ~ l [ 2 , 6 ] , w h i c h a r e c o n s t a n t s for a Fermi-surface

ot l = (l + 1 ) R [ c f / g ] _ l _ 1 + 1 R [ c f / g ] l

(1)

fll = R [ c f / g ] - l - 1

(2)

- R [ c f / g ] l + ( 2 / + 1)

w h e r e c f and g a r e p h y s i c a l l y a l l o w e d s o l u t i o n s of t h e r a d i a l D i r a c e q u a t i o n s t a k e n f o r r = R a n d the energy E = E F = Fermi-energy. In t h e n o n r e l a t i v i s t i c c a s e fil -" 0 a n d ffl ~ ( 2 / + 1 ) R u ; / u l , w h e r e ul(R, E ) / u l ( R , E) i s t h e l o g a r i t h m i c d e r i v a t i v e of t h e (1) s o l u t i o n t o t h e S c h r S d i n g e r e q u a -

159

Volume 30A, n u m b e r 3

PHYSICS

tion for energy EF, evaluated at the radius R. A n o n - z e r o El i s c a u s e d s o l e l y b y s p i n - o r b i t c o u pling. W i t h a c o n s t a n t E s e a r c h we c a l c u l a t e d f o u r Oil, l = 0, 1, 2 a n d 3, b y f i t t i n g f o u r t e e n s e l e c t e d p o i n t s of t h e F e r m i - s u r f a c e [1]. T h e m e a n a b s o l u t e d e v i a t i o n = 0 . 0 0 2 5 FL a n d t h e l a r g e s t a b s o l u t e d e v i a t i o n = 0 . 0 0 6 FL. W i t h t h e s e a l w e c h e c k e d a n o t h e r 12 p o i n t s w h i c h r e s u l t e d in a m e a n a b s o l u t e d e v i a t i o n of 0.003 FL a n d l a r g e s t a b s o l u t e d e v i a t i o n of 0 . 0 1 0 FL. A l s o w e c a l c u l a t e d 15 e x t r e m a l c r o s s - s e c t i o n a l a r e a s in o r d e r to c o m p a r e t h e m w i t h dHvA d a t a [4]. T h e m e a n r e l a t i v e d e v i a t i o n = 1.5% a n d t h e l a r g e s t r e l a t i v e d e v i a t i o n = 3.5%. T h e a g r e e m e n t w i t h e x p e r i ments is very good and proves the remarkable s t r e n g t h of t h e m u f f i n - t i n m o d e l . T h i s i s e s p e c i a l l y s o w h e n on r e a l i z e s t h a t in t h e c a s e of w h i t e t i n 46% of t h e c r y s t a l v o l u m e i s o u t s i d e t h e Slater spheres. We compared our calculations with a local p o t e n t i a l c a l c u l a t i o n [3], w h i c h f i t s v e r y w e l l t h e d H v A e x p e r i m e n t s of [ 3 , 4 ] ( m e a n r e l a t i v e d e v i a t i o n = 1% a n d l a r g e s t r e l a t i v e d e v i a t i o n = 2% on t h e s a m e 15 e x t r e m a l c r o s s - s e c t i o n a l a r e a s ) , b u t s h o w s e r r o r s a s l a r g e a s 0.04 FL a s c o m p a r e d w i t h t h e G a n t m a k h e r e x p e r i m e n t s of [1] ( m e a n a b s o l u t e d e v i a t i o n = 0.012 FL m,d l a r g e s t a b s o l u t e d e v i a t i o n = 0 . 0 4 0 I"L o n t h e a b o v e 14 points). It w a s f o u n d t h a t / 3 1 w a s t h e o n l y i m p o r t a n t spin-orbit parameter. T h e i n f l u e n c e of E2 w a s n e g l i g i b l e . T a k i n g E1 = 0 . 2 0 i m p r o v e d t h e a g r e e m e n t w i t h t h e m e a s u r e d c r o s s - s e c t i o n a l a r e a s of t h e s i x t h z o n e a r o u n d W, w h i c h i s t h e o n l y p l a c e of t h e F e r m i - s u r f a c e that is rather strongly inf l u e n c e d b y s p i n - o r b i t c o u p l i n g , f r o m a b o u t 12 to 3%. It w a s f o u n d t h a t t h e otl h a d v e r y s m a l l w e i g h t s f o r l > / 3 ; m o r e o v e r , d u e to t h e c e n t r i f u g a l t e r m l(l + 1 ) / r 2 i n t h e r a d i a l S c h r ~ l i n g e r equation, the a I approach the empty potential v a l u e s f o r h i g h e r v a l u e s of l. In o u r f i t t i n g p r o cedure we took the lowest four al as variable p a r a m e t e r s a n d a 4 a n d a 5 w e r e t a k e n e q u a l to t h e e m p t y p o t e n t i a l v a l u e s . D u e to n e g l i g i b l e w e i g h t of t h e a l f o r t h e h i g h e r 1 w e n e g l e c t e d t h e m f o r l >t 6. T h e t h r e e l o w e s t a I w e r e v e r y stable for changes in Fermi-surface input data a n d f o r t h e i n t r o d u c t i o n of a 3 a n d E1 a s f u r t h e r parameters. T h e l i m i t a t i o n of t h e n u m b e r of r e c i p r o c a l l a t t i c e v e c t o r s t o a b o u t 120 h a d n e i t h e r a n i n f l u e n c e o n t h e q u a l i t y of t h e f i t n o r o n t h e ~l v a l u e s . T h e s a m e i s t r u e w h e n w e n e g l e c t t h e interaction among the states with the higher z e r o t h o r d e r e n e r g i e s , k. 2 > 7 h 2 / 2 m a 2. Z

160

LETTERS

6 October

Table

1969

1

Best Screened Direct fit atomic pod i r e c t fit tential

EF/h2/2ma2 )

1.60

1.40

1.40

ol0

-7.15

-7.23

-7.22

o~1/3

-1.480

-1.480

-1.453

~2/5

2.05

1.71

1.66

~3/7

3.52

2.07

1.81

fll

0.20

0.24

0.26

m e a n absolute deviation (10 -3 FL)

2.5

2.5

3

l a r g e s t absolute deviation (10 -3 FL)

6

6

6

In t h e c o u r s e of o u r c o m p u t e r e x p e r i m e n t s we discovered that the Fermi-energy EF, def i n e d w i t h r e s p e c t to t h e z e r o - p o t e n t i a l b e t w e e n the Slater spheres, was a very weak parameter; a c h a n g e of E F d i d n o t c h a n g e a p p r e c i a b l y t h e r . m . s , d e v i a t i o n of m e a s u r e d a n d c a l c u l a t e d c a l i p e r s on t h e F e r m i - s u r f a c e . We have chosen for the above calculations, somewhat arbitrarily, E F = 1 . 6 0 h 2 / 2 m a 2 ( s e e t a b l e 3, c o l u m n 1). H o w e v e r , E F = 1.40 o r 1.80 g i v e s f i t s w h i c h a r e a s good. F r o m t h e A P W - m a t r i x o n e c a n s e e t h a t there should be a linear relation between E F and t h e ~ l ' s if t h e y a r e r e q u i r e d to r e p r o d u c e t h e same Fermi-surface in f i r s t o r d e r . S i n g l e A P W wave functions can give a spherical Fermisurface only; off-diagonal matrix elements are e s s e n t i a l f o r a n a c c u r a t e d e s c r i p t i o n of t h e F e r m i - s u r f a c e of a p o l y - v a l e n t m e t a l . T h e a b o v e linear relation states only that the a l (EF) and E F a r e of c o m p a r a b l e i m p o r t a n c e . The fact that the Fermi-surface is independe n t of a r a t h e r l a r g e c h a n g e in E F , a s f o r i n s t a n c e s h o w n b y t h e c o n s t a n c y of t h e t h i r d a n d s i x t h z o n e w i t h r e s p e c t to E F , i n d i c a t e s t h a t probably something more sophisticated than can be taken from the APW-matrix scheme is involved. W e t r i e d to f i n d a n e f f e c t i v e m u f f i n - t i n p o t e n tial by simply screening the atomic HartreeF o c k p o t e n t i a l [5] w i t h e - k r a n d a d d i n g a c o n s t a n t c. T h i s p o t e n t i a l w a s u s e d i n t h e c o u p l e d D i r a c e q u a t i o n s [6]. W i t h o n l y t h e s e two p a r a m e t e r s , k a n d c , we h a v e t r i e d to f i t t h e s a m e fourteen points. We found a rather sharp dep e n d e n c e of t h e f i t o n t h e c h o i c e of E F a n d a b e s t f i t f o r E F = 1.40 h 2 / 2 m a 2 ; w i t h a m e a n a b s o l u t e d e v i a t i o n of 0 . 0 0 3 F L a n d a l a r g e s t d e v i a t i o n of 0.006 FL~ W e f o u n d ~ = 0.1406 (a.u. -1) a n d

Volume 30A, number 3

c = 0.299 Ry. T h e a l ' s , so obtained, a g r e e v e r y w e l l with t h o s e obtained f r o m the d i r e c t fit f o r E F = 1.40 h 2 / 2 m a 2 and/31 = 0.24 (see table 1).

References

SUPERCONDUCTING DISORDERED

3. J . E . Craven, Thesis, Chicago (1968). 4. J . E . Craven and R.W. Stark, Phys. Rev. 168 (1968) 849. 5. F. Herman and S. Skillman, Atomic structure calculations (Prentice Hall, Inc., 1963). 6. T.L. Loucks, Augmented plane wave method (W. A~ Benjamin, Inc., 1967).

* * * * *

1. M.M.M.P. Matthey, Thesis, Nijmegen (1969). 2. L. F. Mattheiss, Phys. Rev. 151 (1966) 450.

THE

6 October 1969

PHYSICS LETTERS

TRANSITION Nb, W, AND

TEMPERATURES MO F I L M S *

OF

J . E. CROW, M. STRONGIN, R.S. THOMPSON and O. F. K A M M E R E R Brookhaven National Laboratory, Upton, New York 11973, USA Received 28 August 1969

Large changes in Tc are observed in films of Nb, W and Mo evaporated under ultrahigh vacuum conditions onto cryogenic substrates. An explanation of these temperature changes in terms of the smearing of N(0) due to the small mean free path is discussed.

It has b e e n known that v a r i a t i o n s of T c o c c u r in f i l m s of t r a n s i t i o n m e t a l s [1-6]. If one c o n s i d e r s the d en s i t y of s t a t e s at the F e r m i l e v e l , N(0), then bib, V and Ta, which have f i v e e l e c t r o n s p e r a t o m , a r e on d e n s i t y of s t a t e s p e a k s and have r e l a t i v e l y high bulk T c ' s [7]. H o w e v e r , W, Mo and Re have r e l a t i v e l y low N ( 0 ) ' s and t h e i r bulk T c ' s a r e low [7]. In d i r t y f i l m s [8] Nb, V and T a, which had l a r g e bulk N ( 0 ) ' s , have d e c r e a s e d T c ' S , and W, Mo and Re, with r e l a t i v e ly s m a l l bulk N(0 )' s , have i n c r e a s e d T c ' s . It is known that g a s e o u s [3-5, 9] i m p u r i t i e s have an effect on the T c ' s of t h e s e m e t a l s , and the e x p l a n a t i o n s f o r the T c c h a n g e s in t h e s e f i l m s have i n v o l v e d g a s e o u s s t a b i l i z e d p h a s e s , compounds and g r a i n s i z e . We find that in the ' d i s o r d e r e d ' f i l m s d e p o s ited onto s u b s t r a t e s at liquid h e l i u m t e m p e r a t u r e the T c of Nb is g r e a t l y r e d u c e d while the T c ' S of W, Mo and Re a r e i n c r e a s e d . As the t e m p e r a t u r e i s i n c r e a s e d th e f i l m s anneal (even below liquid n i t r o g e n t e m p e r a t u r e s in s o m e c a s e s ) , and f o r bib T c goes up, and f o r W and Mo T c g o e s down. The p r e s s u r e during e v a p o r a t i o n w a s 2 × 10 -8 to 6 × 10 -8 T o r r , and a r e s i d u a l gas a n a l y z e r i n d i cat ed an O2 p r e s s u r e l e s s than 10 -9 T o r r . T h e m a j o r i m p u r i t y w a s H 2 followed * This work was performed under the auspices of the US Atomic Energy Commission.

by H 2 0 and A. At 10 -9 T o r r only one m o n o l a y e r of gas can r e a c h the s u b s t r a t e in 40 m i n u t e s . Our e v a p o r a t i o n t i m e s a r e about 5 0 ~ / m i n , and s i n c e De Sorbo [9] has shown than in Nb, 02 r e d u c e s T c at the r a t e of about l O K / a t m % and I-I2 has a n e g l i g i b l e effect, we conclude that the e f f e c t s of r e s i d u a l g a s e s should be n eg l i g i b l e. A f t e r the f i l m is m e a s u r e d in the a s - d e p o s i t e d s t a t e , it is kept u n d er high v a c u u m and is a l lowed to anneal so i t s r e s i s t a n c e changes, and then a new T c point i s taken. Fig. 1 shows data f o r Nb f i l m s . Th e fact that T c d o e s not e x t r a p o l a t e to the usual balk v al u e may be due to a d s o r p t i o n of 0 2 o v e r the t h r e e day annealing p e riod. Molybdenum f i l m s p r e p a r e d on c r y o g e n i c s u b s t r a t e s at a p r e s s u r e of about 3 × 10 -8 T o r r during e v a p o r a t i o n have T c ' s n e a r 6.7OK in the edge r e g i o n s of the f i l m s and n e a r about 4OK when the ed g es a r e m ask ed . A n n eal i n g to r o o m t e m p e r a t u r e r e d u c e s T c below 2.5OK which is the l o w e s t t e m p e r a t u r e we can attain. Tungsten f i l m s p r e p a r e d at ~ 3 × 10 -8 T o r r onto c r y o g e n i c s u b s t r a t e s have a T c n e a r 3OK and an n eal i n g r e d u c e s T c below 2.5OK." In the t r a n s i t i o n m e t a l s the s t r u c t u r e in the d e n s i t y of s t a t e s is of the o r d e r o f 0.1 eV. N i o b i u m , V and T a have t h e i r F e r m i l e v e l s at a peak in the d e n s i t y of s t a t e s , w h i l e W, and Mo a r e e s s e n t i a l l y at v a l l e y s . Th e a s - d e p o s i t e d 161