Determination of the Fermi velocity and cyclotron mass

Volume 20, n u m b e r 2

PHYSICS

LETTERS

References

The ferroelectric transition in potassium dihydrogen phosphate involves a phase change from t e t r a g o n a l to o r t h o r h o m b i c [8], B e l o w t h e t r a n s i t i o n c r e m a i n s a l m o s t c o n s t a n t a t 5 × 104 f o r a l most one degree and then drops gradually. Above t h e t r a n s i t i o n ~ = 3.3 × 1 0 3 / ( T - 1 2 1 . 0 6 2 ) . T h e s e h i g h a n d low t e m p e r a t u r e c u r v e s e x t r a p o l a t e to an intersection point (transition temperature) at 121.127°K. Between 121.127°K and 121.1770K hysteresis in e was observed, thereby showing t h e p o t a s s i u m d i h y d r o g e n p h o s p h a t e t r a n s i t i o n to be f i r s t r a t h e r t h a n s e c o n d o r d e r . A f u l l d e s c r i p t i o n of t h e s e r e s u l t s , i n c l u d i n g t h e e f f e c t of d - c biasing, is in preparation. T h i s i n v e s t i g a t i o n a r o s e f r o m a r e m a r k of D r . M. B l u m e . T h e c o n t i n u i n g i n t e r e s t a n d a d v i c e of D r . B l u m e a n d D r . R. N a t h a n s i s g r a t e f u l l y a c k n o w l e d g e d . D r s . G. S h i r a n e a n d C. F r a z e r p r o v i d e d g u i d a n c e on s a m p l e s e l e c t i o n a n d on c l a s s i c a l m o d e l s of f e , ' r o e l e c t r i c s .

DETERMINATION

OF

THE

FERMI

1 F e b r u a r y 1966

1. C.Domb, Statistical m e c h a n i c s of c r i t i c a l behaviour in magnetic s y s t e m s , in: Magnetism, eds. G. T. Rado and H. Suhl (Academic P r e s s , New York, 1965) Vol. IIa, Ch. 1. See also: P r o c . Intern. Conf. on P h e n o m e n a in the Neighbourhood of C r i t i c a l Points, National Bureau of Standards, Washington, D . C . , April 1965, to be published. 2. C. Domb and M. F. Sykes, J. Math. Phys. 2 (1961) 63; Phys. Rev. 128 (1962) 168. 3. G . A . B a k e r , Phys. Rev. 124 (1961) 768; 129 (1963} 99. 4. M . E . F i s h e r , J . M a t h . P h y s . 4 (1963) 124; 5 (1964) 944. 5. J. E. Noakes and A. A r r o t t , J. Appl. Phys. 35 (1964) 931. 6. J . S . Kouvel and M. E. F i s h e r , Phys. Rev. 126 (1964} A 1626. 7. T e m p e r a t u r e - its m e a s u r e m e n t and control in science and industry, ed. F. G. Brickwedde (Reinhold, New York, 1962) Vol. HI. 8. F. Jona and G. Shirane, F e r r o e l e c t r i c c r y s t a l s ( P e r gamon P r e s s , New York, 1962). 9. S. T r i e b w a s s e r , IBM J. R e s e a r c h Development 1 (1958) 212. 10. J . J . Brophy and S. L. Webb, Phys. Rev. 128 (1962} 584.

VELOCITY

AND

CYCLOTRON

MASS

*

Y. E C K S T E I N

Argonne National Laboratory, Argonne, Illinois Received 3 J a n u a r y 1966

The F e r m i velocity, v F, and cyclotron m a s s , rn*, of m e t a l s is d e t e r m i n e d point by point o v e r the e n t i r e F e r m i s u r f a c e by o b s e r v i n g absorption edges in the magnetoacoustic attenuation which occur when the sound wave is not p e r p e n d i c u l a r to the magnetic field.

To o u r k n o w l e d g e , t h i s i s t h e f i r s t d i r e c t m e a s u r e m e n t of t h e F e r m i v e l o c i t y of a n y m e t a l . The effect has been observed in anitmony and ars e n i c , b u t t h e m e t h o d i s n o t l i m i t e d to s e m i m e t a l s . F i g . 1 i s a n e x a m p l e of t h e o b s e r v e d a t t e n u a t i o n . A t h i g h f i e l d s t h e a t t e n u a t i o n i s v e r y low, a n d a s t h e f i e l d i s d e c r e a s e d , a s e r i e s of s h a r p increases in the attenuation (absorption edges) occur in pairs. It i s p o s s i b l e to u n d e r s t a n d a b s o r p t i o n e d g e s by considering Doppler-shifted cyclotron reson a n c e [1]: An e l e c t r o n w i l l a b s o r b s o u n d r e s o n a n t ly o n l y if t h e e f f e c t i v e s o u n d f r e q u e n c y O)eff ( w h i c h * B a s e d on work p e r f o r m e d under the auspices of the U. S. Atomic E n e r g y C o m m i s s i o n . 142

i s t h e D o p p l e r - s h i f t e d f r e q u e n c y ) i s e q u a l to a n i n t e g r a l m u l t i p l e of t h e c y c l o t r o n f r e q u e n c y w c. Thus, for resonant absorption,

OJeff = ~ [ 1 - ~ . ( v > / v s ]

= n~ c ,

(1)

where w is the sound frequency, (v) the timeaverage electron velocity, ~ a unit vector in the d i r e c t i o n of t h e s o u n d w a v e v e c t o r q, v , t h e s o u n d v e l o c i t y , a n d n a n i n t e g e r . E q . (1~ i s a l s o the condition for energy-momentum conservation in the electron-phonon interaction; n is then the difference in electron Landau levels before and a f t e r phonon a b s o r p t i o n . Since wc i s p r o p o r t i o n a l to t h e m a g n e t i c f i e l d a n d t h e v e l o c i t y ( v ) i s bounded, a maximum field for given n exists above which there is no resonant absorption. For a con-

Volume 20, number 2

PHYSICS

LETTERS

1 F e b r u a r y 1966

v e x s u r f a c e it c a n be s h o w n t h a t the m a x i m u m f i e l d o c c u r s f o r e l e c t r o n s at the e l l i p t i c l i m i t i n g p o i n t of t h e F e r m i s u r f a c e , w h e r e the e l e c t r o n v e l o c i t y i s p a r a l l e l to the m a g n e t i c f i e l d ¢, If H + i s the m a x i m u m f i e l d f o r p o s i t i v e n, and Hn- the m a x i m u m f i e l d f o r n e g a t i v e n, t h e n (1) g i v e s u s m )k

vF

=

he(H;

m*

=

ne(H[ -

vr = (q--:m

+

gn)/[2c(q.~i) ] ,

(2a)

Hj)l[2cm]

(2b)

,

(2c)

~o

Ioo

I~o

zoo

250

300

MAGNETIC FIELD (Gouss)

w h e r e the a b s o l u t e v a l u e of n i s u n d e r s t o o d , and v F and m * a r e the l i m i t i n g p o i n t v a l u e s . In t h i s e x p e r i m e n t , l o n g i t u d i n a l s o u n d w a v e s of f r e q u e n c i e s up to 344 M c / s w e r e p r o p a g a t e d a l o n g the b i n a r y a x i s of a s i n g l e c r y s t a l of a n t i m o n y . A m a g n e t i c f i e l d of 0 - 3 0 0 G w a s r o t a t e d in the b i n a r y - b i s e c t r i x and b i n a r y - t r i g o n a l p l a n e s . The e x p e r i m e n t a l s e t u p i s i d e n t i c a l w i t h t h a t u s e d f o r the c o n v e n t i o n a l m a g n e t o - a c o u s t i c o s c i l l a t i o n m e a s u r e m e n t s , e x c e p t t h a t the s p e c i m e n h o l d e r c a n be r o t a t e d in the p l a n e of q and H, r a t h e r t h a n p e r p e n d i c u l a r to q. A b s o r p t i o n e d g e s w e r e c l e a r l y o b s e r v e d f o r m a g n e t i c f i e l d s at a n g l e s f r o m the p a r a l l e l to the s o u n d w a v e d i r e c t i o n to 30 ° f r o m it. The s h a r p r i s e s in a t t e n u a t i o n in fig. 1 axe i n t e r p r e t e d a s due to the o n s e t of the a b s o r p t i o n e d g e s . The s u b s e q u e n t d e c r e a s e in the a t t e n u a t i o n at m a g n e t i c f i e l d s b e l o w the p e a k f i e l d f o r A, B and C i s due to g e o m e t r i c r e s o n a n c e o s c i l l a t i o n s , a s d i s c u s s e d in the f o l l o w i n g p a p e r [2]. T h e s e p e a k s a r e the f i r s t m a x i m a in g e o m e t r i c r e s o n a n c e s f o r the t r a n s i t i o n s n = ~- 1 (peaks A and ]3) and f o r n = + 2 (peak C). The o t h e r g e o m e t r i c r e s o n a n c e m a x i m a f o r n = + 1 and n = + 2 w e r e not o b s e r v e d , and a r e b e l i e v e d to b e b e l o w the n o i s e l e v e l . The s h a r p p e a k i n g f a c i l i t a t e d o b s e r v a t i o n of the s p l i t t i n g of the e d g e s f o r ± n, and t h u s m a d e p o s s i b l e a m e a s u r e m e n t of the F e r m i velocity. The p e a k s A and B c a n n o t be i n t e r p r e t e d a s a s e r i e s of the u s u a l g e o m e t r i c r e s o n a n c e o s c i l l a t i o n s (n = 0 t r a n s i t i o n s ) b e c a u s e t h e y a r e not p e r i o d i c in l/H; a l s o t h e y o c c u r o n l y in p a i r s w h i c h m o v e r e l a t i v e to e a c h o t h e r a s the f i e l d a n g l e i s c h a n g e d . In a d d i t i o n , the p e a k s c a n n o t be i n t e r p r e t e d a s r e s o n a n c e s s i m i l a r to t h o s e ob¢ This differs from the Pz lira defined in [6] and [7]. Our mterpretatmn of magneto-acoustm effects for oblique geometry differs from Kaner, but is in a g r e e ment with Kjeldaas [1] for qll ti"

Fig. 1. ql] binary; till 6 ° from binary in b i n a r y - b i s e c trix plane: f= 265 M c / s . The peaks A 1 and A2 a r e interpreted as first maxima in the attenuation f6r absorption edges with n = -~1. The peaks B 1 and B 2 are f i r s t maxima of absorption edges with n = ~1 due to another sheet of the F e r m i surface; C 1 and C 2 are subharmonies of A 1 and A 2 and correspond tb edges-with n = ~_2. At low fields, below 100 G, geometric resonances from the n = 0 transitions are observed [6]. s e r v e d by J o n e s [3] in A1, b e c a u s e the F e r m i s u r f a c e of a n t i m o n y is c l o s e l y e l l i p s o i d a l . F r o m fig. 1 we find f o r the p a i r A1, A2: m * v F = 0.91 X 10 - 2 1 g c m / s e c , m * = 0.084 too, and v F = 1.19 × 107 c m / s e c ; f o r the p a i r B1, B2: m * v F = 0.76 x 10-21 g c r a / s e c , m * = 0.074 too, and v F = 1.13 × 107 c m / s e c . The e s t i m a t e d e r r o r in m * v F is 5% due to a m b i g u i t y in d e t e r m i n i n g the p o s i t i o n of the e d g e . The e f f e c t i v e m a s s w a s c a l c u l a t e d by u s i n g the d i f f e r e n c e in p e a k f i e l d v a l u e s (which s h o u l d be a l m o s t e q u a l to the diff e r e n c e in the a b s o r p t i o n e d g e f i e l d v a l u e s ) and the m a j o r e r r o r (about 5%) i s due to the s u b t r a c t i o n of two l a r g e n u m b e r s . T h e s e v a l u e s a r e c o n s i s t e n t w i t h the d a t a on e l e c t r o n s in a n t i m o n y [4, 5], c o r r e s p o n d i n g to the two r o t a t e d e l l i p s o i d s . We w o u l d l i k e to thank J. B. K e t t e r s o n f o r interesting discussions.

References 1. 2. 3. 4.

T. Kjeldaas, Phys. Rev. 113 (1959} 1473. S.G. Eckstein, following letter. B.K. Jones, Phil. Mag. 9 (1964) 217. L. Windmiller and M. G. PriesHey, Solid State Comm.3 (1965} 199. 5. Y. Eckstein, J . B . Ketterson and S. G. Eckstein, Phys. Hey. 135A (1964) 740. 6. E . A . Kaner, V.D. Peschanskii and I. A. Privorotskii, Zh. Eksperim. i T e o r . F i z . 40 (1961) 214; Soviet P h y s . - J E T P 13 (1961) 147. 7. A. P. Korolyak and L. Ya. Matsakov, Pisma v Redaktsiyu 2 (1965) 30; J E T P Letters 2 (1965) 18.

143