Asymptotic behavior of electrical and hall conductivities of tin

Volume24A, number 5

PHYSICS L E T T E R S

4. S.Masunaga, I. Morita and M. Ishiguro, J. Phys. Soc. Japan 21 {1966) 638. 5. H.Kanzaki and K. Kido, J. Phys. Soc. Japan 15

2. Y.Toyozawa and M.Inoue, J. Phys. Soc. Japan 21 (1966) 1663. 3. A. Fukuda, Sci. Light (Tokyo) 13 (1964) 64; T. Mabuchi, A. Fukuda and R. Onaka, Sci. Light

(1960) 529. 6. S.Sugano, J. Chem. Phys. 36 (1962) 122.

(Tokyo) 15 (1966) 79.

ASYMPTOTIC

27 February 1967

BEHAVIOR OF ELECTRICAL CONDUCTIVITIES OF TIN *

AND

HALL

K. C. BORDOLOI

University of Louisville, Louisville, Kentucky and C. G. GRENIER and J . M. REYNOLDS

Louisiana State University, Baton Rouge, Louisiana Received 7 January 1967

The asymptotic behavior of electrical and Hall conductivities of tin is associated with magnetic breakdown. Gold and Priestley's free-electron model of the F e rm i surface of tin is taken as a basis for comparison with our experimental findings.

A band a n a l y s i s p e r f o r m e d on the g r o s s - e f f e c t s of a l l and a12 at 4.2°K, b a s e d on a f o u r band S o n d h e i m e r - W i l s o n m o d e l , r e s u l t e d in good a g r e e m e n t when the e f f e c t i v e m a s s e s w e r e c o m puted and c o m p a r e d with t h o s e obtained f r o m Gold and P r i e s t l e y ' s [1] f r e e - e l e c t r o n m o d e l . The a s y m p t o t i c b e h a v i o r o b s e r v e d in ~12 in high fiel d can be u n d e r s t o o d f r o m the zone 4(a) of that m o d e l . The c a r r i e r s in this zone have e l e c t r o n b e h a v i o r (the o r b i t ~) i n s t e a d of hole b e h a v i o r . Th e total n u m b e r of s t a t e s in zone 4(a) contained in the c y l i n d e r s Oe4(a)) and between the c o r r u g a t e d p l a n e s (nh4(b)) a m o u n t s a p p r o x i m a t e l y to a s l i c e of the B r i l l o u i n zone of t h i c k n e s s 2ApZ ~ 2~rti/a (a/c - 2c/a) and will contain about 0.40 e l e c t r o n p e r a to m , in good a g r e e m e n t with ou r e x p e r i m e n t a l r e s u l t of 0.34 e l e c t r o n p e r a tom. T he c o n s t a n t t e r m of ~11 (/4=00) was a s s o c i a ted with the e f f e c t of m a g n e t i c breakdown [2-3]. It m a y be noted that if the s t r u c t u r e f a c t o r along the XL line (fig. 1) i s z e r o , then the e n e r g y gap a c r o s s the B r i l l o u i n p l a n e along the XL line would be e x p e c t e d to be v a n i s h i n g l y s m a l l . Then the o r b i t c o r r e s p o n d i n g to zone 4(a) s y s t e m m a y * This work was supported by U.S. Atomic Energy Commission.

shift f r o m the o r b i t ~ to the f r e a k o r b i t (i). Such a situation is b e t t e r known as m a g n e t i c b r e a k down w h e r e the B r a g g r e f l e c t i o n may fail if the condition ~¢o >> ~ A 2 / E F

(1)

i s r e a c h e d [4]. H e r e a ~- 1, A is the e n e r g y gap. A s s u m i n g that the e n e r g y gap is s m a l l along the XL line (Pz = 0) and has the s m a l l v al u e A = A o at the i n t e r c e p t of the XL line with the f r e e - e l e c t r o n F e r m i s p h e r e , then at a n o t h e r point off the b a s a l plane (Pz ¢ 0), it would be e x p e c t e d that A would i n c r e a s e as Pz i s i n c r e a s e d . Letting A= = A o + Bpz 2, eq. (1) b e c o m e s :

(ehHEF/rncot) ½ --
(2)

F o r s m a l l v a l u e s of A , a s m a l l s l i c e around the b a s a l plane may be su b j ect to the m a g n e t i c b r e a k down p r o b a b i l i t y with a t h i c k n e s s 2~pz such that 1 1 x _1 )~ APz = (ehEF / B 2 m c a ) ~ (H 2 - H g _ (3) with

e h H / m c = (~ Ao2 / E F.

F o r this s m a l l s l i c e of t h i c k n e s s 2 5 p z the m o s t p r o b a b l e si t u at i o n would be as shown in the l a r g e s t zone (zone 3 + zone 4a) with the f r e a k hole o r b i t (i). T h i s would h a v e the effect of i n 269

Volume 24A, n u m b e r 5

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PHYSICS

LETTERS

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References

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c r e a s i n g t h e n u m b e r of h o l e s b y t h e a m o u n t (2c/a) ~Pz/Po h o l e s p e r a t o m ; a n a m o u n t w h i c h i s a relatively small quantity with almost no field d e p e n d e n c e . H e n c e , i t w o u l d b e d i f f i c u l t to a s s e r t i t s p r e s e n c e i n t h e a12 b e h a v i o r . B u t t h e s u p p o s i t i o n of c o m p l e t e b r e a k d o w n i n t h e s l i c e of z o n e of t h i c k n e s s 2~P z i s o n l y a n o v e r s i m p l i f i c a t i o n , and it would be expected that a partial breakdown s h o u l d t a k e p l a c e , m o s t l y in t h e u p p e r a n d t h e l o w e r p a r t s of t h e s l i c e . T h e o r b i t s a r e t h e n a r a n d o m s u c c e s s i o n of p a r t s of o r b i t s ~ a n d ~, t h u s c a u s i n g a r a n d o m w a l k of t h e e l e c t r o n s [5], t h i s r a n d o m m o t i o n b e i n g e q u i v a l e n t to p a r t l y compensated hole and electron orbits and open orbits. The breakdown situation would then lead t o a n a l m o s t f i e l d i n d e p e n d e n t t e r m in t h e a l l c o e f f i c i e n t . If we a s s i g n a n e f f e c t i v e m a s s m = m o to t h e r a d o m l y w a l k i n g e l e c t r o n s , m a g n e t i c b r e a k d o w n s h o u l d a f f e c t a p p r o x i m a t e l y 0.01 e l e c t r o n p e r a t o m to e x p l a i n t h e a s y m p t o t i c b e h a v i o r of Crli.

I I

~___.,,k. ! 7, __, ~f I_

27 F e b r u a r y 1967

I,,,

Fig. 1. The basal c r o s s section of the F e r m i s u r f a c e of tin in the extended zone s c h e m e based on Gold and P r i e s t l e y ' s f r e e - e l e c t r o n model.

NORMAL

VIBRATIONS

J. BERC~MA,

1. A.V. Gold and M. G. P r i e s t l e y , Phil. Mag. 5 (1960) 1089. 2. K. Bordoloi, C.G. G r e n i e r and J . M . R e y n o l d s , Bull. Am. Phys. Soc. 10 (1965)127. 3. R.C.Young, Phys. Rev. L e t t e r s 15 0965) 262. 4. J . M . Ziman, T h e o r y of Solids (Cambridge U n i v e r s i ty P r e s s , Cambridge, 1964). 5. A . B . Pippard, P r o c . Roy. Soc. {London)A 270 (1962) 1.

IN

at-IRON*

C. V A N D I J K a n d D. T O C C H E T T I

Reactor Centrun7 Nederland, Petten, The Netherlands Received 6 F e b r u a r y 1967

Detailed m e a s u r e m e n t s of phonon d i s p e r s i o n r e l a t i o n s in the s y m m e t r y d i r e c t i o n s of at-iron by m e a n s of inelastic neutron s c a t t e r i n g a r e r e p o r t e d . A n a l y s i s on the b a s i s of the B o r n - yon K~rm~u theory shows that f o r c e s out to fifth neighbours a r e significant.

T h e l a t t i c e d y n a m i c s of b c c m e t a l s h a s b e e n discussed by several authors and different at* Work sponsored jointly by R e a c t o r C e n t r u m N e d e r land, The Netherlands and Instituut for Atomenergi, Norway. 270

tempts have been made to take into account the eff e c t of t h e c o n d u c t i o n e l e c t r o n s [1]. F o r t h e c a s e of a t - i r o n i n c l u s i o n of t h i s e f f e c t i s r a t h e r d i f f i cult due to its complicated electronic structure. M o r e o v e r , t h e e x p e r i m e n t a l d a t a of t h e p h o n o n