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De Haas-van Alphen effect and Fermi surface in niobium
Volume 27A, number 10
PHYSICS LETTERS
7 October 1968
t e c h n i q u e d e v e l o p e d by the B r u s s e l s School. O t h e r r e l e v a n t work in this d i r e c t i o n i s d i s c u s s e d and r e f e r e n c e d in [7] and [8]. The p r e s e n t viewpoint can be i l l u s t r a t e d by studying the e l e m e n t a r y e x a m p l e of sect i o n 3B of r e f . 3, in which it is found a p p r o p r i a t e to employ f o r m u l a (1). One now finds that the f i r s t - o r d e r - i n - ~ c a l c u l a tion of G3(x , x) w can be i n t e r p r e t e d in f a m i l i a r t e r m s , n a m e l y :
G3(x, x)¢o =- (1/21r) 2 ([go(-W)" [ g o ( - 2 w ) ( - L i )go(-2co)x;x];x]) , w h e r e go(-W) --- ( L o + W ) - i , and H = H o + HI, with H o e p 2 / 2 m + ½mw2x 2 and H I ~ _ ~ 3. Evaluation of this e x p r e s s i o n f o r G3(x , x) w i n v o l v e s b i n o m i a l expansion of go, r e s u l t i n g in p r o d u c t s of s i m p l e s e r i e s which can be e a s i l y w r i t t e n in c l o s e d f o r m [3, eq. (34)]. T he author i s indebted to K. C. Hines f o r h is i n t e r e s t in this work.
R e f e~F~c e s
1. 2. 3. 4. 5. 6. 7. 8.
E.R. Pike, Proc. Phys. Soe. (London) 84 (1964) 83. J.C. Herzel, J. Math. Phys. 8 (1967) 1650. T. Tanaka, K. Moorjani and T. Morita, Phys. Rev. 155 0967) 388. N.N. Bogolyubov Jr. and B.I. Sadovnikov, Zh. Eksperim. i Teor. Fiz. 43 (1962) 677; Soviet Phys. JETP 16 (1963) 482. T.Kawasaki, Prog. Theor. Phys. (Kyoto) 39 (1968} 331. A.Ron, J. Math. Phys. 4 (1963} 811. S.A. Rice and P. Gray, The statistical mechanics of simple liquids (Interscience Publishers, New York, 1965} especially section 7.3.E. P. R~sibois, in Physics of many-particle systems, Vol. 1. Ed. E. Meeron (Gordon and Breach, Science Publishers, Inc., New York, 1966} Chap. 6 section 8.
DE
HAAS-VAN
ALPHEN
EFFECT
AND
FERMI
SURFACE
IN NIOBIUM
t
G. B. SCOTT, M. SPRINGFORD and J . R. STOCKTON
School of Mathematical and Physical Sciences, University of Sussex, Brighton, UK Received 26 August 1968
De Haas-van Alphen oscillations have been investigated in niobium and are discussed in terms of a model for the Fermi surface proposed by Mattheiss.
We have studied de H a a s - v a n Alphen (dHvA) o s c i l l a t i o n s in niobium $ f o r m a g n e t i c f i e l d d i r e c tions in the { 110} p l a n e s . T h e r e s u l t s a r e i n t e r p r e t e d in t e r m s of a m o d e l f o r the F e r m i s u r f a c e p r o p o s e d by M a t t h e i s s [1] f o r the Group VB t r a n sition m e t a l s but s o m e f e a t u r e s of this m o d e l a r e not o b s e r v e d in our e x p e r i m e n t s . T h e dHvA o s c i l lations w e r e i n v e s t i g a t e d in a 100 kG s u p e r c o n t This work is supported by the Science Research Council. We are indebted to Professor W. F. Vinen, Department of Physics, University of Birmingham, and to Dr. G. Taylor, Department of Metallurgy, University of Oxford, for the niobium crystals.
ducting m a g n e t using the f i el d modulation t e c h nique [2]. Fig. 1 shows the o b s e r v e d a n g u l a r v a r i a t i o n of e x t r e m a l a r e a in the ( l I 0 ) plane. T h e high o _ f r e q u e n c y b r a n c h has a m i n i m u m a r e a of 1.85A -~ with m * / m o = 3.2 along [111] and extends f o r about 20 ° on e i t h e r s i d e of the [111] o r i e n t a t i o n . This is in a g r e e m e n t with m a g n e t o t h e r m a l e x p e r i m e n t s of G r a e b n e r et al. [3] who have s u g g e s t e d that this b r a n c h d e r i v e s f r o m e x t r e m a l o r b i t s on t h e open hole sh eet about the H point in the Bril]ouin zone, and r e c e n t c a l c u l a t i o n s by M a t t h e i s s [4] a g r e e quantitively with the p r o p o s a l . Fig. 1 a l s o shows a set of dHvA f r e q u e n c i e s which
655
Volume 27A, number 10
I
I
1
]1--
PHYSICS
I
I
I
I
I
•
Z'0-
1'9--
% 7o - -
1.9
.,,=S:
~ 0.~ &A •AU,
•&
~l,& & & •
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0-7F
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0"5
•••
• •••
• =AAAA&•AA•
A•••••~,••••
""
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Oqk==~,•• • dd~, 6 • I I [100]
] 0
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L 117Lll
Fig. 1. Extremal c r o s s sectional areas of the F e r m i surface in niobium in units of/~-2, as a function of magnetic field orientation in the (li0) plane. The c r y s tal rotation axis was 1° off [1~0] and in consequence consists of two frequencies which differ by ~ 2% over part of the range. l i e in the r a n g e 0.6A -2 to g r e a t e r than 0.8,~ -2 with m*/m o in t h e r a n g e 1.4 to 1.6. V a l u e s f o r t h e s e at (1107 h a v e b e e n g i v e n by T h o r s e n and B e r l i n c o u r t [5] and a r e r e p o r t e d in m a g n e t o t h e r m a l e x p e r i m e n t s [6]. T h e s y m m e t r y and d e g e n e r a c y of t h e b r a n c h e s i s c o n s i s t e n t with a s s i g n i n g t h e m to d i s t o r t e d e l l i p s o d i a l p o c k e t s s i t u a t e d at N in t h e B r i l l o u i n z o n e and h a v i n g s e m i - a x i s d i mensions in NP~ N F and NH d i r e c t i o n s of 0.57, 0.48 and 0 . 4 2 A - " r e s p e c t i v e l y . B e c a u s e t h e a x e s N F and NH of any one of t h e e l l i p s o i d s a r e r e s p e c t i v e l y a l w a y s p a r a l l e l to t h e a x e s NH and N F
65G
LETTERS
7 October 1968
of one of t h e o t h e r f i v e e l l i p s o i d s in t h e B r i l l o u i n z o n e , it is not p o s s i b l e e x p e r i m e n t a l l y to a s s i g n t h e two c o r r e s p o n d i n g s e m i - a x i s v a l u e s . W e h a v e t h e r e f o r e u s e d the r e s u l t of t h e A P W c a l c u l a t i o n [1] w h i c h i s that N F > NH. E x t e n d i n g f o r about 30 ° in the ( l i 0 ) p l a n e , two low f r e q u e n c y b r a n c h e s h a v e b e e n d e t e c t e d with m i n i m u m a r e a s of 0 . 0 8 A -2 and 0.14A -2 f o r t h e m a g n e t i c f i e l d a l o n g [100]. m * / m o f o r t h e f o r m e r h a s b e e n m e a s u r e d a s 0.29. It i s r e a s o n a b l e to a s s o c i a t e t h e s e in t e r m s of the p r e s e n t m o d e l with e x t r e m a l o r b i t s a r o u n d the (100) - d i r e c t e d a r m s of the open h o l e s h e e t . T h e s e m a y be c o m p a r e d with an e s t i m a t e f o r t h e d i m e n s i o n s of t h e a r m s d e d u c e d by F a w c e t t et al. [7] who f r o m H a l l e f f e c t m e a s u r e m e n t s find t h e m i n i m u m d i m e n s i o n of the a r m s to b e 021 FH in the ~110} p l a n e s . W i t h t h e a s s u m p t i o n of a c i r c u l a r c r o s s s e c t i o n p e r p e n d i c u l a r to FH, t h i s y i e l d s 0.096A -2 f o r the m i n i m u m c r o s s s e c t i o n a l a r e a s n o r m a l to (100)$, a v a l u e w h i c h l i e s in t h e r a n g e of the p r e s e n t m e a s u r e m e n t s . F r o m t h e s y m m e t r y of t h e 1-]-I a r m s , only an odd n u m b e r of e x t r e m a l o r b i t s can e x i s t about e a c h of the (1007 d i r e c t i o n s . It i s s u g g e s t e d that the f r e q u e n c i e s m a y d e r i v e f r o m u n d u l a t i n g ( 1 0 0 7 - d i r e c t e d a r m s and f u r t h e r e x p e r i m e n t s m o r e s u i t e d to the o b s e r v a t i o n of t h e s e low f r e q u e n c i e s m a y r e v e a l a d d i t i o n s t r u c t u r e . N o t i c e a b l y a b s e n t f r o m fig. 1 i s a f r e q u e n c y b r a n c h which could be a s s o c i a t e d with t h e s u g g e s t e d [1] c l o s e d h o l e ' o c t a h e d r a l ' s u r f a c e c e n t r e at r . T h e d i m e n s i o n s of t h i s s u r f a c e g i v e n by M a t t h e i s s [1] w i l l be m o d i f i e d by s p i n - o r b i t i n t e r a c t i o n , a l t h o u g h t h e s e e f f e c t s a r e not e x p e c t e d to be l a r g e in n i o b i u m . ~[ The lat~tice parameter for niobium has been taken as 3.3008A.
References 1. L. F. Mattheiss, Phys. Rev. 139 (1965) A1893. 2. D. Shoenberg and P. J. Stiles, P r e c . Roy. Soc. (London) A281 (1964) 62. 3. J . E . Graebner, J.H. Condon, F . S . L . Hsu and J . E . Kunzler, Bull. Am. Phys. Soc. 13 (1968) 508. 4. L. F. Mattheiss, Bull. Am. Phys. Soc. 13 (1968) 508. 5. A. C. Thorsen and T. G. Berlincourt, Phys. Rev. Letters 7 (1961) 244. 6. M. G. Halloran, F . S . L . Hsu and J. E. Kunzler, Bull. Am. Phys. Soc. 13 (1968) 59. 7. E. Fawcett, W.A. Reed and R. R. Soden, Phys. Rev. 159 (1967) 533.
PHYSICS LETTERS
7 October 1968
t e c h n i q u e d e v e l o p e d by the B r u s s e l s School. O t h e r r e l e v a n t work in this d i r e c t i o n i s d i s c u s s e d and r e f e r e n c e d in [7] and [8]. The p r e s e n t viewpoint can be i l l u s t r a t e d by studying the e l e m e n t a r y e x a m p l e of sect i o n 3B of r e f . 3, in which it is found a p p r o p r i a t e to employ f o r m u l a (1). One now finds that the f i r s t - o r d e r - i n - ~ c a l c u l a tion of G3(x , x) w can be i n t e r p r e t e d in f a m i l i a r t e r m s , n a m e l y :
G3(x, x)¢o =- (1/21r) 2 ([go(-W)" [ g o ( - 2 w ) ( - L i )go(-2co)x;x];x]) , w h e r e go(-W) --- ( L o + W ) - i , and H = H o + HI, with H o e p 2 / 2 m + ½mw2x 2 and H I ~ _ ~ 3. Evaluation of this e x p r e s s i o n f o r G3(x , x) w i n v o l v e s b i n o m i a l expansion of go, r e s u l t i n g in p r o d u c t s of s i m p l e s e r i e s which can be e a s i l y w r i t t e n in c l o s e d f o r m [3, eq. (34)]. T he author i s indebted to K. C. Hines f o r h is i n t e r e s t in this work.
R e f e~F~c e s
1. 2. 3. 4. 5. 6. 7. 8.
E.R. Pike, Proc. Phys. Soe. (London) 84 (1964) 83. J.C. Herzel, J. Math. Phys. 8 (1967) 1650. T. Tanaka, K. Moorjani and T. Morita, Phys. Rev. 155 0967) 388. N.N. Bogolyubov Jr. and B.I. Sadovnikov, Zh. Eksperim. i Teor. Fiz. 43 (1962) 677; Soviet Phys. JETP 16 (1963) 482. T.Kawasaki, Prog. Theor. Phys. (Kyoto) 39 (1968} 331. A.Ron, J. Math. Phys. 4 (1963} 811. S.A. Rice and P. Gray, The statistical mechanics of simple liquids (Interscience Publishers, New York, 1965} especially section 7.3.E. P. R~sibois, in Physics of many-particle systems, Vol. 1. Ed. E. Meeron (Gordon and Breach, Science Publishers, Inc., New York, 1966} Chap. 6 section 8.
DE
HAAS-VAN
ALPHEN
EFFECT
AND
FERMI
SURFACE
IN NIOBIUM
t
G. B. SCOTT, M. SPRINGFORD and J . R. STOCKTON
School of Mathematical and Physical Sciences, University of Sussex, Brighton, UK Received 26 August 1968
De Haas-van Alphen oscillations have been investigated in niobium and are discussed in terms of a model for the Fermi surface proposed by Mattheiss.
We have studied de H a a s - v a n Alphen (dHvA) o s c i l l a t i o n s in niobium $ f o r m a g n e t i c f i e l d d i r e c tions in the { 110} p l a n e s . T h e r e s u l t s a r e i n t e r p r e t e d in t e r m s of a m o d e l f o r the F e r m i s u r f a c e p r o p o s e d by M a t t h e i s s [1] f o r the Group VB t r a n sition m e t a l s but s o m e f e a t u r e s of this m o d e l a r e not o b s e r v e d in our e x p e r i m e n t s . T h e dHvA o s c i l lations w e r e i n v e s t i g a t e d in a 100 kG s u p e r c o n t This work is supported by the Science Research Council. We are indebted to Professor W. F. Vinen, Department of Physics, University of Birmingham, and to Dr. G. Taylor, Department of Metallurgy, University of Oxford, for the niobium crystals.
ducting m a g n e t using the f i el d modulation t e c h nique [2]. Fig. 1 shows the o b s e r v e d a n g u l a r v a r i a t i o n of e x t r e m a l a r e a in the ( l I 0 ) plane. T h e high o _ f r e q u e n c y b r a n c h has a m i n i m u m a r e a of 1.85A -~ with m * / m o = 3.2 along [111] and extends f o r about 20 ° on e i t h e r s i d e of the [111] o r i e n t a t i o n . This is in a g r e e m e n t with m a g n e t o t h e r m a l e x p e r i m e n t s of G r a e b n e r et al. [3] who have s u g g e s t e d that this b r a n c h d e r i v e s f r o m e x t r e m a l o r b i t s on t h e open hole sh eet about the H point in the Bril]ouin zone, and r e c e n t c a l c u l a t i o n s by M a t t h e i s s [4] a g r e e quantitively with the p r o p o s a l . Fig. 1 a l s o shows a set of dHvA f r e q u e n c i e s which
655
Volume 27A, number 10
I
I
1
]1--
PHYSICS
I
I
I
I
I
•
Z'0-
1'9--
% 7o - -
1.9
.,,=S:
~ 0.~ &A •AU,
•&
~l,& & & •
&•
0-7F
••
t •
0"5
•••
• •••
• =AAAA&•AA•
A•••••~,••••
""
0.Z"
Oqk==~,•• • dd~, 6 • I I [100]
] 0
l
i
] [111]
I
L 117Lll
Fig. 1. Extremal c r o s s sectional areas of the F e r m i surface in niobium in units of/~-2, as a function of magnetic field orientation in the (li0) plane. The c r y s tal rotation axis was 1° off [1~0] and in consequence consists of two frequencies which differ by ~ 2% over part of the range. l i e in the r a n g e 0.6A -2 to g r e a t e r than 0.8,~ -2 with m*/m o in t h e r a n g e 1.4 to 1.6. V a l u e s f o r t h e s e at (1107 h a v e b e e n g i v e n by T h o r s e n and B e r l i n c o u r t [5] and a r e r e p o r t e d in m a g n e t o t h e r m a l e x p e r i m e n t s [6]. T h e s y m m e t r y and d e g e n e r a c y of t h e b r a n c h e s i s c o n s i s t e n t with a s s i g n i n g t h e m to d i s t o r t e d e l l i p s o d i a l p o c k e t s s i t u a t e d at N in t h e B r i l l o u i n z o n e and h a v i n g s e m i - a x i s d i mensions in NP~ N F and NH d i r e c t i o n s of 0.57, 0.48 and 0 . 4 2 A - " r e s p e c t i v e l y . B e c a u s e t h e a x e s N F and NH of any one of t h e e l l i p s o i d s a r e r e s p e c t i v e l y a l w a y s p a r a l l e l to t h e a x e s NH and N F
65G
LETTERS
7 October 1968
of one of t h e o t h e r f i v e e l l i p s o i d s in t h e B r i l l o u i n z o n e , it is not p o s s i b l e e x p e r i m e n t a l l y to a s s i g n t h e two c o r r e s p o n d i n g s e m i - a x i s v a l u e s . W e h a v e t h e r e f o r e u s e d the r e s u l t of t h e A P W c a l c u l a t i o n [1] w h i c h i s that N F > NH. E x t e n d i n g f o r about 30 ° in the ( l i 0 ) p l a n e , two low f r e q u e n c y b r a n c h e s h a v e b e e n d e t e c t e d with m i n i m u m a r e a s of 0 . 0 8 A -2 and 0.14A -2 f o r t h e m a g n e t i c f i e l d a l o n g [100]. m * / m o f o r t h e f o r m e r h a s b e e n m e a s u r e d a s 0.29. It i s r e a s o n a b l e to a s s o c i a t e t h e s e in t e r m s of the p r e s e n t m o d e l with e x t r e m a l o r b i t s a r o u n d the (100) - d i r e c t e d a r m s of the open h o l e s h e e t . T h e s e m a y be c o m p a r e d with an e s t i m a t e f o r t h e d i m e n s i o n s of t h e a r m s d e d u c e d by F a w c e t t et al. [7] who f r o m H a l l e f f e c t m e a s u r e m e n t s find t h e m i n i m u m d i m e n s i o n of the a r m s to b e 021 FH in the ~110} p l a n e s . W i t h t h e a s s u m p t i o n of a c i r c u l a r c r o s s s e c t i o n p e r p e n d i c u l a r to FH, t h i s y i e l d s 0.096A -2 f o r the m i n i m u m c r o s s s e c t i o n a l a r e a s n o r m a l to (100)$, a v a l u e w h i c h l i e s in t h e r a n g e of the p r e s e n t m e a s u r e m e n t s . F r o m t h e s y m m e t r y of t h e 1-]-I a r m s , only an odd n u m b e r of e x t r e m a l o r b i t s can e x i s t about e a c h of the (1007 d i r e c t i o n s . It i s s u g g e s t e d that the f r e q u e n c i e s m a y d e r i v e f r o m u n d u l a t i n g ( 1 0 0 7 - d i r e c t e d a r m s and f u r t h e r e x p e r i m e n t s m o r e s u i t e d to the o b s e r v a t i o n of t h e s e low f r e q u e n c i e s m a y r e v e a l a d d i t i o n s t r u c t u r e . N o t i c e a b l y a b s e n t f r o m fig. 1 i s a f r e q u e n c y b r a n c h which could be a s s o c i a t e d with t h e s u g g e s t e d [1] c l o s e d h o l e ' o c t a h e d r a l ' s u r f a c e c e n t r e at r . T h e d i m e n s i o n s of t h i s s u r f a c e g i v e n by M a t t h e i s s [1] w i l l be m o d i f i e d by s p i n - o r b i t i n t e r a c t i o n , a l t h o u g h t h e s e e f f e c t s a r e not e x p e c t e d to be l a r g e in n i o b i u m . ~[ The lat~tice parameter for niobium has been taken as 3.3008A.
References 1. L. F. Mattheiss, Phys. Rev. 139 (1965) A1893. 2. D. Shoenberg and P. J. Stiles, P r e c . Roy. Soc. (London) A281 (1964) 62. 3. J . E . Graebner, J.H. Condon, F . S . L . Hsu and J . E . Kunzler, Bull. Am. Phys. Soc. 13 (1968) 508. 4. L. F. Mattheiss, Bull. Am. Phys. Soc. 13 (1968) 508. 5. A. C. Thorsen and T. G. Berlincourt, Phys. Rev. Letters 7 (1961) 244. 6. M. G. Halloran, F . S . L . Hsu and J. E. Kunzler, Bull. Am. Phys. Soc. 13 (1968) 59. 7. E. Fawcett, W.A. Reed and R. R. Soden, Phys. Rev. 159 (1967) 533.