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Magneto-acoustic oscillations in thallium
Volume 2, number 3
I~HYSICS
MAGNETO-ACOUSTIC
LETTERS
1 Septe]nber 1962
OSCILLATIONS
IN
THALLIUM
J. A. RAYNE Westinghouse Research Laboratories, Pittsburgh, Pennsylvania Received 9 August 1962
The b a n d s t r u c t u r e of t h a l l i u m i s of c o n s i d e r a b l e t h e o r e t i c a l i n t e r e s t , s i n c e it i s the o n l y t r i v a l e n t met~tl h a v i n g a h e x a g o n a l c l o s e - p a c k e d s t r u c t u r e a n d the e f f e c t s of s p i n - o r b i t s p l i t t i n g a r e p r e s u m a b l y l a r g e . H i t h e r t o t h e r e has b e e n a v a i l a b l e n o i n formation concerning its Fermi surface, apart from s o m e f r a g m e n t a r y d a t a 1) o b t a i n e d f r o m l o w - f i e l d De H a a s - V a n A l p h e n m e a s u r e m e n t s . In t h i s l e t t e r we r e p o r t f o r the f i r s t t i m e , the o b s e r v a t i o n of 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 s in t h a l l i u m . The r e s u l t i n g d i m e n s i o n s of t h e F e r m i s u r f a c e a r e c o m p a r e d with the p r e d i c t i o n s of t h e f r e e e l e c t r o n m o d e l . In g e n e r a l the a g r e e m e n t i s found to be q u i t e good if a s i n g l e zone s c h e m e i s u s e d . T h u s the i m p o r t a n c e of s p i n - o r b i t e f f e c t s on t h e b a n d s t r u c t u r e of m e t a l s h a v i n g high a t o m i c n u m b e r i s conclusively demonstrated. A s i n g l e c r y s t a l of t h a l l i u m w a s p r o d u c e d b y a s t r a i n a n n e a l t e c h n i q u e u s i n g 99.999+% p u r e s t a r t i n g m a t e r i a l o b t a i n e d f r o m the A m e r i c a n S m e l t I n g a n d R e f i n i n g C o m p a n y . The r e s u l t i n g ingot w a s o r i e n t e d by X - r a y s and s u i t a b l e a c o u s t i c s p e c i m e n s cut f r o m it b y a s p a r k e r o s i o n a p p a r a t u s . T r a n s m i s s i o n m e a s u r e m e n t s were made at frequencies up to 270 M c / s u s i n g an a u t o m a t i c r e c o r d i n g t e c h n i q u e d e s c r i b e d p r e v i o u s l y 2). In a l l c a s e s l o n g i t u d i n a l s o u n d w a v e s w e r e u s e d . Fig. 1 shows the t r a c i n g of a t y p i c a l s e t of d a t a with q If [0001] f o r two d i f f e r e n t f i e l d d i r e c t i o n s . F r o m t h e p e r i o d s of t h e o b s e r v e d o s c i l l a t i o n s , the c o r r e s p o n d i n g e x t r e m a l d i m e n s i o n s of the F e r m i s u r f a c e in the d i r e c t i o n q x H w e r e c o m p u t e d f r o m the f o r m u l a e k kext = ~-c A(1/H) "
(1)
H e r e ), i s the sound w a v e l e n g t h o b t a i n e d f r o m the m e a s u r e d e l a s t i c c o n s t a n t s a n d A(1/H) the a p p r o p r i a t e p e r i o d . The r e s u l t i n g d i m e n s i o n s a r e s u m m a r i s e d in t a b l e 1. F o r a l l f i e l d d i r e c t i o n s t h e r e a p p e a r to b e t h r e e d i s t i n c t p e r i o d s of o s c i l l a t i o n . O n l y the s h o r t p e r i o d i s w e l l d e f i n e d in a l l c a s e s ; t h e i n t e r m e d i a t e p e r i o d i s m u c h c l e a r e r f o r H II [1010], while the l o n g e r p e r i o d i s m o r e d i s t i n c t f o r H [] [11~.0]. A l l t h r e e s e t s of e x t r e m a l d i m e n s i o n s a r e , h o w e v e r , i s o t r o p i c within the a c c u r a c y of the p r e s e n t data. 128
I
O9
I I
I
I --
I 3
--
I
i 4
IO00/H (oersted-I)
Fig. 1. Oscillatory attenuation of longitudinal 269 Mc/s sound wave in thaUtu.m as a function of 1/H. The propagation direction is [0001], while the m a g netic field is in the basal plane. It i s of i n t e r e s t to c o m p a r e the e x p e r i m e n t a l F e r m i s u r f a c e d i m e n s i o n s with t h o s e o b t a i n e d f r o m the f r e e e l e c t r o n m o d e l . H a r r i s o n 3) h a s t r e a t e d the c a s e of a t r i v a l e n t h e x a g o n a l c l o s e - p a c k e d m e t a l a s s u m i n g a double zone s c h e m e . A c t u a l l y the e f f e c t s of s p i n - o r b i t s p l i t t i n g a r e e x p e c t e d to b e q u i t e l a r g e
Table 1 Comparison of extremal dimensions for the F e r m i surface of T1 with free electron model.
Propagation direction
Field direction
Experimental
[0001]
[11~0]
1.68 0.64 0.14 1.46 0.52 0.14 0.62 1.46 0.62
[10i0]
[11~:0]
[1i00]
[10~0]
[1210]
[0001] [0001]
kex t * Free electron value
**
1.38 - 1.52 0.38 - 0.52 ? 1.76 - 2.10 0.88 0.54 1.38 0.54 1.76
-
0.86 1.52 0.86 2.10
* All units of 108 cm -1 . ** The range of values corresponds to the possible extrema of the free electron model.
Volume 2, number 3
PHYSICS
in t h a l l i u m , r e s u l t i n g in an a p p r e c i a b l e gap a c r o s s t h e (0001) f a c e s e x c e p t in t h e [1010] d i r e c t i o n w h e r e t h e r e i s an a c c i d e n t a l d e g e n e r a c y . The r e s u l t i n g s i n g l e - z o n e s c h e m e h a s f i r s t and s e c o n d bands which are full, the third and fourth bands b e i n g shown s c h e m a t i c a l l y in f i g . 2. T h e r e a r e s m a l l p o c k e t s of e l e c t r o n s on t h e fifth a n d s i x t h z o n e s , but t h e s h a p e o r even t h e p r e s e n c e of t h e s e regions for a finite crystal potential is somewhat doubtful. R e f e r e n c e to t a b l e 1 s h o w s t h e r e i s r e a s o n a b l e a g r e e m e n t b e t w e e n t h e d i m e n s i o n s of t h e f r e e e l e c t r o n m o d e l a n d t h o s e o b t a i n e d e x p e r i m e n t a l l y . The short period oscillation is identifiable with orbits a r o u n d t h e h e x a g o n a l c u p s f o r m e d b y the e x t e n d e d t h i r d zone s u r f a c e . One half of s u c h an o r b i t i s shown b y t h e d o t t e d c u r v e ~ of f i g . 2. F o r q [] [0001] t h e e x t r e m a l d i m e n s i o n i s in t h e b a s a l p l a n e ; the o b s e r v e d i s o t r o p y of t h e s h o r t p e r i o d m u s t t h e r e f o r e i m p l y t h a t t h e i n t e r i o r c o r n e r s of t h e t h i r d z o n e s u r f a c e a r e c o n s i d e r a b l y r o u n d e d . It i s a l s o of i n t e r e s t t h a t , f o r t h i s d i r e c t i o n of p r o p a g a t i o n , t h e r e w a s no o b s e r v a b l e e f f e c t w h i c h c o u l d b e a s s o c i a t e d w i t h m a g n e t i c b r e a k d o w n a c r o s s t h e (0001) zone f a c e s . T h e o s c i l l a t i o n s of t h e i n t e r m e d i a t e p e r i o d a r e p r e s u m a b l y a s s o c i a t e d with o r b i t s s u c h a s B, w h i c h e n c o m p a s s t h e p r i s m o i d a l s u r f a c e s a l o n g t h e [ 1010} f a c e s . F o r / / II [11~01 t h e e x t r e m a l o r b i t s would b e b e t t e r d e f i n e d than f o r / 1 I[ [10~0], t h u s a c c o u n t i n g f o r t h e g r e a t e r a m p l i t u d e of o s c i l l a t i o n in t h e f o r m e r c a s e . It i s b e l i e v e d t h a t t h e l o n g p e r i o d o s c i l l a t i o n s a r e a s s o c i a t e d with o r b i t s i n v o l v i n g t h e f i f t h a n d s i x t h zone s u r f a c e s . A d e t a i l e d c o m p a r i s o n of t h e s e d i m e n s i o n s with t h e f r e e e l e c t r o n m o d e l d o e s not seem possible at this juncture. Magneto-acoustic oscillations have also been o b s e r v e d with q II [ i 0 1 0 ] and q II [ i 1 2 0 ] . F o r / I II [0001] p r o n o u n c e d s h o r t p e r i o d o s c i l l a t i o n s o c c u r , w h i l e f o r / / i n t h e b a s a l p l a n e the p e r i o d s a r e m u c h l o n g e r . A s m a y b e s e e n f r o m t a b l e 1, t h e l a t t e r c o r r e s p o n d to e x t r e m a l v a l u e s of k w h i c h a r e in q u i t e good a g r e e m e n t with t h e c e n t r a l d i m e n s i o n s , m e a s u r e d a l o n g [0001], of o r b i t s s u c h a s c~. T h i s w o u l d a p p e a r to b e s t r o n g e v i d e n c e f o r t h e c o r r e c t n e s s of t h e s i n g l e zone m o d e l and h e n c e f o r t h e e x i s t e n c e of a s t r o n g s p i n - o r b i t s p l i t t i n g . F o r q If [ I 0 1 0 ] , / / II [1210], i t m i g h t b e a n t i c i p a t e d t h a t t h e e x t r e m a l d i m e n s i o n would c o r r e s p o n d to a d o u b l e z o n e o r b i t o w i n g to t h e a c c i d e n t a l b a n d d e g e n e r a c y along [i010]. Unless the crystal orientation were
LETTERS
1 September 1962
3rd Zone
//~,.
[I 1-20]
Fig. 2. Cross sections of the f r e e - e l e c t r o n F e r m i surface for thallium assuming a single zone scheme. The f i r s t two zones a r e full. Small pockets in the fifth and sixth zones have been omitted. v e r y a c c u r a t e , h o w e v e r , t h e b a n d gap would be s u f f i c i e n t f o r t h i s e f f e c t to b e u n o b s e r v a b l e . T h a n k s a r e due to M r . P . H o d d e r and D r . P . A . F l i n n f o r t h e i r a i d in p r e p a r i n g t h e s i n g l e c r y s t a l of t h a l l i u m u s e d in t h e s e e x p e r i m e n t s . T h e a u t h o r would a l s o l i k e to a c k n o w l e d g e v a l u a b l e d i s c u s s i o n s with D r s . W . A . H a r r i s o n , L. F a l i c o v and M. P r i e s t ley. 1) D. Shoenberg, Phil. Trans. Roy. Soc. A 245 (1952) 1. 2) J . A . R a y n e and H.Sell, Phys. Rev. Letters 8 (1962) 199. 3) W . A . H a r r i s o n , Phys. Rev. 118 (1960) 1190.
129
I~HYSICS
MAGNETO-ACOUSTIC
LETTERS
1 Septe]nber 1962
OSCILLATIONS
IN
THALLIUM
J. A. RAYNE Westinghouse Research Laboratories, Pittsburgh, Pennsylvania Received 9 August 1962
The b a n d s t r u c t u r e of t h a l l i u m i s of c o n s i d e r a b l e t h e o r e t i c a l i n t e r e s t , s i n c e it i s the o n l y t r i v a l e n t met~tl h a v i n g a h e x a g o n a l c l o s e - p a c k e d s t r u c t u r e a n d the e f f e c t s of s p i n - o r b i t s p l i t t i n g a r e p r e s u m a b l y l a r g e . H i t h e r t o t h e r e has b e e n a v a i l a b l e n o i n formation concerning its Fermi surface, apart from s o m e f r a g m e n t a r y d a t a 1) o b t a i n e d f r o m l o w - f i e l d De H a a s - V a n A l p h e n m e a s u r e m e n t s . In t h i s l e t t e r we r e p o r t f o r the f i r s t t i m e , the o b s e r v a t i o n of 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 s in t h a l l i u m . The r e s u l t i n g d i m e n s i o n s of t h e F e r m i s u r f a c e a r e c o m p a r e d with the p r e d i c t i o n s of t h e f r e e e l e c t r o n m o d e l . In g e n e r a l the a g r e e m e n t i s found to be q u i t e good if a s i n g l e zone s c h e m e i s u s e d . T h u s the i m p o r t a n c e of s p i n - o r b i t e f f e c t s on t h e b a n d s t r u c t u r e of m e t a l s h a v i n g high a t o m i c n u m b e r i s conclusively demonstrated. A s i n g l e c r y s t a l of t h a l l i u m w a s p r o d u c e d b y a s t r a i n a n n e a l t e c h n i q u e u s i n g 99.999+% p u r e s t a r t i n g m a t e r i a l o b t a i n e d f r o m the A m e r i c a n S m e l t I n g a n d R e f i n i n g C o m p a n y . The r e s u l t i n g ingot w a s o r i e n t e d by X - r a y s and s u i t a b l e a c o u s t i c s p e c i m e n s cut f r o m it b y a s p a r k e r o s i o n a p p a r a t u s . T r a n s m i s s i o n m e a s u r e m e n t s were made at frequencies up to 270 M c / s u s i n g an a u t o m a t i c r e c o r d i n g t e c h n i q u e d e s c r i b e d p r e v i o u s l y 2). In a l l c a s e s l o n g i t u d i n a l s o u n d w a v e s w e r e u s e d . Fig. 1 shows the t r a c i n g of a t y p i c a l s e t of d a t a with q If [0001] f o r two d i f f e r e n t f i e l d d i r e c t i o n s . F r o m t h e p e r i o d s of t h e o b s e r v e d o s c i l l a t i o n s , the c o r r e s p o n d i n g e x t r e m a l d i m e n s i o n s of the F e r m i s u r f a c e in the d i r e c t i o n q x H w e r e c o m p u t e d f r o m the f o r m u l a e k kext = ~-c A(1/H) "
(1)
H e r e ), i s the sound w a v e l e n g t h o b t a i n e d f r o m the m e a s u r e d e l a s t i c c o n s t a n t s a n d A(1/H) the a p p r o p r i a t e p e r i o d . The r e s u l t i n g d i m e n s i o n s a r e s u m m a r i s e d in t a b l e 1. F o r a l l f i e l d d i r e c t i o n s t h e r e a p p e a r to b e t h r e e d i s t i n c t p e r i o d s of o s c i l l a t i o n . O n l y the s h o r t p e r i o d i s w e l l d e f i n e d in a l l c a s e s ; t h e i n t e r m e d i a t e p e r i o d i s m u c h c l e a r e r f o r H II [1010], while the l o n g e r p e r i o d i s m o r e d i s t i n c t f o r H [] [11~.0]. A l l t h r e e s e t s of e x t r e m a l d i m e n s i o n s a r e , h o w e v e r , i s o t r o p i c within the a c c u r a c y of the p r e s e n t data. 128
I
O9
I I
I
I --
I 3
--
I
i 4
IO00/H (oersted-I)
Fig. 1. Oscillatory attenuation of longitudinal 269 Mc/s sound wave in thaUtu.m as a function of 1/H. The propagation direction is [0001], while the m a g netic field is in the basal plane. It i s of i n t e r e s t to c o m p a r e the e x p e r i m e n t a l F e r m i s u r f a c e d i m e n s i o n s with t h o s e o b t a i n e d f r o m the f r e e e l e c t r o n m o d e l . H a r r i s o n 3) h a s t r e a t e d the c a s e of a t r i v a l e n t h e x a g o n a l c l o s e - p a c k e d m e t a l a s s u m i n g a double zone s c h e m e . A c t u a l l y the e f f e c t s of s p i n - o r b i t s p l i t t i n g a r e e x p e c t e d to b e q u i t e l a r g e
Table 1 Comparison of extremal dimensions for the F e r m i surface of T1 with free electron model.
Propagation direction
Field direction
Experimental
[0001]
[11~0]
1.68 0.64 0.14 1.46 0.52 0.14 0.62 1.46 0.62
[10i0]
[11~:0]
[1i00]
[10~0]
[1210]
[0001] [0001]
kex t * Free electron value
**
1.38 - 1.52 0.38 - 0.52 ? 1.76 - 2.10 0.88 0.54 1.38 0.54 1.76
-
0.86 1.52 0.86 2.10
* All units of 108 cm -1 . ** The range of values corresponds to the possible extrema of the free electron model.
Volume 2, number 3
PHYSICS
in t h a l l i u m , r e s u l t i n g in an a p p r e c i a b l e gap a c r o s s t h e (0001) f a c e s e x c e p t in t h e [1010] d i r e c t i o n w h e r e t h e r e i s an a c c i d e n t a l d e g e n e r a c y . The r e s u l t i n g s i n g l e - z o n e s c h e m e h a s f i r s t and s e c o n d bands which are full, the third and fourth bands b e i n g shown s c h e m a t i c a l l y in f i g . 2. T h e r e a r e s m a l l p o c k e t s of e l e c t r o n s on t h e fifth a n d s i x t h z o n e s , but t h e s h a p e o r even t h e p r e s e n c e of t h e s e regions for a finite crystal potential is somewhat doubtful. R e f e r e n c e to t a b l e 1 s h o w s t h e r e i s r e a s o n a b l e a g r e e m e n t b e t w e e n t h e d i m e n s i o n s of t h e f r e e e l e c t r o n m o d e l a n d t h o s e o b t a i n e d e x p e r i m e n t a l l y . The short period oscillation is identifiable with orbits a r o u n d t h e h e x a g o n a l c u p s f o r m e d b y the e x t e n d e d t h i r d zone s u r f a c e . One half of s u c h an o r b i t i s shown b y t h e d o t t e d c u r v e ~ of f i g . 2. F o r q [] [0001] t h e e x t r e m a l d i m e n s i o n i s in t h e b a s a l p l a n e ; the o b s e r v e d i s o t r o p y of t h e s h o r t p e r i o d m u s t t h e r e f o r e i m p l y t h a t t h e i n t e r i o r c o r n e r s of t h e t h i r d z o n e s u r f a c e a r e c o n s i d e r a b l y r o u n d e d . It i s a l s o of i n t e r e s t t h a t , f o r t h i s d i r e c t i o n of p r o p a g a t i o n , t h e r e w a s no o b s e r v a b l e e f f e c t w h i c h c o u l d b e a s s o c i a t e d w i t h m a g n e t i c b r e a k d o w n a c r o s s t h e (0001) zone f a c e s . T h e o s c i l l a t i o n s of t h e i n t e r m e d i a t e p e r i o d a r e p r e s u m a b l y a s s o c i a t e d with o r b i t s s u c h a s B, w h i c h e n c o m p a s s t h e p r i s m o i d a l s u r f a c e s a l o n g t h e [ 1010} f a c e s . F o r / / II [11~01 t h e e x t r e m a l o r b i t s would b e b e t t e r d e f i n e d than f o r / 1 I[ [10~0], t h u s a c c o u n t i n g f o r t h e g r e a t e r a m p l i t u d e of o s c i l l a t i o n in t h e f o r m e r c a s e . It i s b e l i e v e d t h a t t h e l o n g p e r i o d o s c i l l a t i o n s a r e a s s o c i a t e d with o r b i t s i n v o l v i n g t h e f i f t h a n d s i x t h zone s u r f a c e s . A d e t a i l e d c o m p a r i s o n of t h e s e d i m e n s i o n s with t h e f r e e e l e c t r o n m o d e l d o e s not seem possible at this juncture. Magneto-acoustic oscillations have also been o b s e r v e d with q II [ i 0 1 0 ] and q II [ i 1 2 0 ] . F o r / I II [0001] p r o n o u n c e d s h o r t p e r i o d o s c i l l a t i o n s o c c u r , w h i l e f o r / / i n t h e b a s a l p l a n e the p e r i o d s a r e m u c h l o n g e r . A s m a y b e s e e n f r o m t a b l e 1, t h e l a t t e r c o r r e s p o n d to e x t r e m a l v a l u e s of k w h i c h a r e in q u i t e good a g r e e m e n t with t h e c e n t r a l d i m e n s i o n s , m e a s u r e d a l o n g [0001], of o r b i t s s u c h a s c~. T h i s w o u l d a p p e a r to b e s t r o n g e v i d e n c e f o r t h e c o r r e c t n e s s of t h e s i n g l e zone m o d e l and h e n c e f o r t h e e x i s t e n c e of a s t r o n g s p i n - o r b i t s p l i t t i n g . F o r q If [ I 0 1 0 ] , / / II [1210], i t m i g h t b e a n t i c i p a t e d t h a t t h e e x t r e m a l d i m e n s i o n would c o r r e s p o n d to a d o u b l e z o n e o r b i t o w i n g to t h e a c c i d e n t a l b a n d d e g e n e r a c y along [i010]. Unless the crystal orientation were
LETTERS
1 September 1962
3rd Zone
//~,.
[I 1-20]
Fig. 2. Cross sections of the f r e e - e l e c t r o n F e r m i surface for thallium assuming a single zone scheme. The f i r s t two zones a r e full. Small pockets in the fifth and sixth zones have been omitted. v e r y a c c u r a t e , h o w e v e r , t h e b a n d gap would be s u f f i c i e n t f o r t h i s e f f e c t to b e u n o b s e r v a b l e . T h a n k s a r e due to M r . P . H o d d e r and D r . P . A . F l i n n f o r t h e i r a i d in p r e p a r i n g t h e s i n g l e c r y s t a l of t h a l l i u m u s e d in t h e s e e x p e r i m e n t s . T h e a u t h o r would a l s o l i k e to a c k n o w l e d g e v a l u a b l e d i s c u s s i o n s with D r s . W . A . H a r r i s o n , L. F a l i c o v and M. P r i e s t ley. 1) D. Shoenberg, Phil. Trans. Roy. Soc. A 245 (1952) 1. 2) J . A . R a y n e and H.Sell, Phys. Rev. Letters 8 (1962) 199. 3) W . A . H a r r i s o n , Phys. Rev. 118 (1960) 1190.
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