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Elastic moduli of Bi2Te3 from 4.2°K to 300°K
Volume 30A, number 6
PHYSICS LE]?TERS
ELASTIC
MODULI
OF Bi2Te 3 FROM
17 November 1969
4.2°K
TO 300°K
J. O. JENKINS, J . A . RAYNE
Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, USA and R. W. URE
Westinghouse Research and Development Center, Pittsburgh, Pennsylvania 15235, USA Received 13 October 1969
The elastic moduli of bismuth telluride have been measured from 4.2°K to 300°K using a continuous wave resonance technique. At 4.2°K the me asuremenks give C] 1 = 7.436, C66 = 2.619, C33 = 5.160, C44 = 3.135 and CI~ = 1.541 in units of 10l ' dyne/cmz. These values correspond to a limiting Debye temperature of 158 K, which agrees well with the calorimetric figure.
B i s m u t h t e l l u r i d e (Bi2Te3) , which belongs to the space group R3m [1], has s i x independent e l a s t i c moduli. This l e t t e r r e p o r t s an u l t r a s o n i c d e t e r m i n a t i o n of these moduli f r o m 4.2°K to 300OK, using a continuous wave r e s o n a n c e m e t h od [2] at 10 MHz. Two s a m p l e s , with the s a m e o r i e n t a t i o n of faces as used in r e c e n t work on antimony [3], were cut f r o m an ingot of n - t y p e m a t e r i a l having a c a r r i e r c o n c e n t r a t i o n of about 3 x 1018 c m -3. The s a m p l e s were a p p r o x i m a t e l y 3 x 6 x 5 m m 3, in size with the longest d i m e n s i o n along the b i n a r y axis. Velocites Vl, v2, v3, v7, v8, v9, V l l , v12 , v13 and v14 [3] were m e a s u r e d , although v2, v 9 and v12 were d e t e r m i n e d only by 300°K. The v a l u e s of the density and l i n e a r dim e n s i o n s of the s a m p l e s were c o r r e c t e d for lattice c o n t r a c t i o n below room t e m p e r a t u r e , using the Gr~ineisen r e l a t i o n and published heat capacity data. T a b l e 1 shows the r e s u l t s of this work at 4.2°K, 80°K and 300°K, together with p r e v i o u s data at 300°K for p-type m a t e r i a l [4]. The a g r e e m e n t is s a t i s f a c t o r y in view of the i m c o m p l e t e n e s s and l o w e r p r e c i s i o n of the l a t t e r m e a s u r e m e n t s . It should be noted that the sign for C14 depends on the choice of C a r t e s i a n r e f e r e n c e axes r e l a t i v e to the c r y s t a l b a s i s v e c t o r s ; the convention used in refs. 3 and 5 has been followed here. The m a i n u n c e r t a i n t y in the p r e s e n t data r e s u l t s f r o m the m e a s u r e m e n t of s a m p l e length and its extrapolation to low t e m p e r a t u r e s . It is e s t i m a t e d that the moduli a r e p r o b a b l y r e l i able to 0.25%, except for C13 which could be d e t e r m i n e d to 0.5%.
rable 1 Elastic moduli of Bi2Te 3 in units of 1011 dyne/cm 2. Modulus
Temperature (OK) 80 300
4.2
Ref. 4
Cll C66 C33 C44 C13
7.436 2.619 5.160 3.135 2.916
7.332 2.570 5.080 3.059 2.874
6.847 2.335 4.768 2.738 2.703
6.46 2.88 4.73 2.50 --
C14
1.541
1.500
1.325
--
The r e q u i r e m e n t of e l a s t i c stability is that the 6 × 6 m a t r i x of the e l a s t i c moduli be positive definite. F o r Bi2Te 3 this condition is equivalent to the r e l a t i o n s : C l l >C66 >0;
C44C66-C24 >0;
C33(Cll-C66)
C23 > 0 .
(1)
T h e s e conditions a r e well s a t i s f i e d by the moduli of table 1, the t h i r d r e l a t i o n being useful in r e j e c t i n g a s p u r i o u s root for C 13". I n t e r l a y e r bonds of the weak van a e r Waals type have been p r o p o s e d to account for the very easy cleavage of Bi2Te 3 and the soft acoustic modes, which c a u s e a d e p a r t u r e f r o m 7~ behavior in the low t e m p e r a t u r e specific heat [6]. The r e l a t i v e i m p o r t a n c e of c e n t r a l f o r c e s is indicated by the extent to which the Cauchy r e l a tions a r e satisfied. F o r t r i g o n a l s t r u c t u r e s these a r e given by: 3C66 = C l l ;
C44 = C 1 3 .
(2) 349
Volume 30A, number 6
PHYSICS
The present data deviate from obeying these e q u a l i t i e s by 5.7 and 7.5 p e r c e n t r e s p e c t i v e l y , s u g g e s t i n g t h a t the e l a s t i c b e h a v i o r is i n d e e d d o m i n a t e d by c e n t r a l van d e r W a a l s f o r c e s . A l a t t i c e d y n a m i c a l c a l c u l a t i o n is c u r r e n t l y b e i n g m a d e to e x p l o i t t h i s s i m p l e b o n d i n g s c h e m e . The limiting Debye temperature, obtained f r o m the m o d u l i at 4.2OK and u s i n g 167.17 ,~3 as the v o l u m e of the p r i m i t i v e c e l l , h a s b e e n c o m p u t e d by d i r e c t n u m e r i c a l i n t e g r a t i o n to be 158 + l ° K . T h i s r e s u l t a g r e e s with the c a l o r i m e t r i c v a l u e of 162 + 3OK, within the c o m b i n e d uncertainties.
LETTERS
17 November 1969
References 1. J . R . Drabble in P r o g r e s s in semiconductors, eds A. F. Gibson and R. E. Burgess, (Heywood and Co., London 1963) Vol. 7, p. 45. 2. D.I. Bolef and M. Menes, J. Appl. Phys. 31 {1960) 1010. 3. S. Epstein and A. P. de Bretteville J r . , Phys. Rev. 138 (1965) A 771. 4. Y. Ililavskii, Soy. Phys. Solid State 3 (1962) 2582. 5. R.B. Mallinson, J . A . Rayne and R. W. Ure J r . , Phys. Rev. 175 (1968) 1049. 6. G. Shoemake, J. Rayne and R. W. Ure J r . , Phys. Rev., to be published.
T h i s w o r k w a s s u p p o r t e d by a g r a n t f r o m the N a t i o n a l S c i e n c e Foundation.
P U L S E D S P I N - P R E C E S S I O N R E - O R I E N T A T I O N M E T H O D TO S T U D Y S P I N E X C H A N G E OF F R E E IONS * H. A. SCHUESSLER** Department of Physics, University of Washington, Seattle, Washington 98105, USA Received 16 October 1969
A scheme of consecutive square r.f. pulses was developed to study the electron spin re-orientation process of paramagnetic ions in spin exchange collisions with polarized atoms.
The high d e g r e e of i s o l a t i o n r e a l i z e d when s t o r i n g i o n s in a r. f. q u a d r u p o l e t r a p [1,2] s h o u l d m a k e a v a r i e t y of a t o m i c c o l l i s i o n e x p e r i m e n t s on c h a r g e d a t o m i c s y s t e m s p o s s i b l e . T h i s l e t t e r r e p o r t s the f i r s t m e a s u r e m e n t of an i o n - a t o m spin exchange time. Stored polarized ions were s u b j e c t e d to a s e q u e n c e of s h o r t r e s o n a n t r.f. p u l s e s , f o r e a c h of w h i c h the p u l s e l e n g t h Td and the p u l s e h e i g h t H 1 w e r e a d j u s t e d to p r o d u c e a 180 ° r o t a t i o n of the e l e c t r o n i c spin p o l a r i z a t i o n . T h e p u l s e p a r a m e t e r s then f u l f i l l the c o n d i t i o n s w 1 T d = z H 1 T d = II, T d < Te. A p o l a r i z e d a t o m i c b e a m p a s s i n g t h r o u g h the t r a p i n t e r a c t e d with the s t o r e d ions. T h e s p i n e x c h a n g e p r o c e s s c h a r a c t e r i z e d by the spin e x c h a n g e t i m e T e p a r t l y r e o r i e n t s the i n v e r t e d e l e c t r o n i c s p i n s of the i o n s d u r i n g the long r.f. f r e e p e r i o d s Tp b e t w e e n the * Supported by the U.S. Army Research Office (Durham). and by the National Science Foundation. ** Present Address: Department of Physics, Texas A and M University, College Station, Texas 77843. 350
s h o r t 180 ° p u l s e s . T e m a y be s t u d i e d by o b s e r v ing the t i m e e v o l u t i o n of the p o l a r i z a t i o n as a f u n c t i o n of T_. The n o v e l P s c h e m e w a s f i r s t u s e d to study the spin e x c h a n g e p r o c e s s CsT + 3He+~ ~ Cs~ + 3He+~
(1)
A l t h o u g h d u r i n g s p i n e x c h a n g e i t s e l f the n u c l e u s is not i m m e d i a t e l y a f f e c t e d , a n g u l a r m o m e n t u m is s u b s e q u e n t l y t r a n s f e r r e d to the n u c l e u s by the hfs i n t e r a c t i o n A I ' S and two e x c h a n g e c o l l i s i o n s a r e n e c e s s a r y to o r i e n t 3He+. The i n t e r e s t i n g Iris Z e e m a n l e v e l s a r e ( 1 , 1 ) ~ (1,0) at Vd, (I,i) ~-~ (i,-I) at re, and (i,0)~-~ (I,-i) at vf. °He+ ions were created inside a r.f. quadrupole trap by,pulsed electron bombardment of the background ~He gas, and the stored ions of an energy of a few eV were subjected to an optically pumped beam of thermal Cs atoms. In the resulting collisions, the ions quickly assumed the polarization of the beam by spin exchange. Their polarization was then inverted in each 180o pulse. The
PHYSICS LE]?TERS
ELASTIC
MODULI
OF Bi2Te 3 FROM
17 November 1969
4.2°K
TO 300°K
J. O. JENKINS, J . A . RAYNE
Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213, USA and R. W. URE
Westinghouse Research and Development Center, Pittsburgh, Pennsylvania 15235, USA Received 13 October 1969
The elastic moduli of bismuth telluride have been measured from 4.2°K to 300°K using a continuous wave resonance technique. At 4.2°K the me asuremenks give C] 1 = 7.436, C66 = 2.619, C33 = 5.160, C44 = 3.135 and CI~ = 1.541 in units of 10l ' dyne/cmz. These values correspond to a limiting Debye temperature of 158 K, which agrees well with the calorimetric figure.
B i s m u t h t e l l u r i d e (Bi2Te3) , which belongs to the space group R3m [1], has s i x independent e l a s t i c moduli. This l e t t e r r e p o r t s an u l t r a s o n i c d e t e r m i n a t i o n of these moduli f r o m 4.2°K to 300OK, using a continuous wave r e s o n a n c e m e t h od [2] at 10 MHz. Two s a m p l e s , with the s a m e o r i e n t a t i o n of faces as used in r e c e n t work on antimony [3], were cut f r o m an ingot of n - t y p e m a t e r i a l having a c a r r i e r c o n c e n t r a t i o n of about 3 x 1018 c m -3. The s a m p l e s were a p p r o x i m a t e l y 3 x 6 x 5 m m 3, in size with the longest d i m e n s i o n along the b i n a r y axis. Velocites Vl, v2, v3, v7, v8, v9, V l l , v12 , v13 and v14 [3] were m e a s u r e d , although v2, v 9 and v12 were d e t e r m i n e d only by 300°K. The v a l u e s of the density and l i n e a r dim e n s i o n s of the s a m p l e s were c o r r e c t e d for lattice c o n t r a c t i o n below room t e m p e r a t u r e , using the Gr~ineisen r e l a t i o n and published heat capacity data. T a b l e 1 shows the r e s u l t s of this work at 4.2°K, 80°K and 300°K, together with p r e v i o u s data at 300°K for p-type m a t e r i a l [4]. The a g r e e m e n t is s a t i s f a c t o r y in view of the i m c o m p l e t e n e s s and l o w e r p r e c i s i o n of the l a t t e r m e a s u r e m e n t s . It should be noted that the sign for C14 depends on the choice of C a r t e s i a n r e f e r e n c e axes r e l a t i v e to the c r y s t a l b a s i s v e c t o r s ; the convention used in refs. 3 and 5 has been followed here. The m a i n u n c e r t a i n t y in the p r e s e n t data r e s u l t s f r o m the m e a s u r e m e n t of s a m p l e length and its extrapolation to low t e m p e r a t u r e s . It is e s t i m a t e d that the moduli a r e p r o b a b l y r e l i able to 0.25%, except for C13 which could be d e t e r m i n e d to 0.5%.
rable 1 Elastic moduli of Bi2Te 3 in units of 1011 dyne/cm 2. Modulus
Temperature (OK) 80 300
4.2
Ref. 4
Cll C66 C33 C44 C13
7.436 2.619 5.160 3.135 2.916
7.332 2.570 5.080 3.059 2.874
6.847 2.335 4.768 2.738 2.703
6.46 2.88 4.73 2.50 --
C14
1.541
1.500
1.325
--
The r e q u i r e m e n t of e l a s t i c stability is that the 6 × 6 m a t r i x of the e l a s t i c moduli be positive definite. F o r Bi2Te 3 this condition is equivalent to the r e l a t i o n s : C l l >C66 >0;
C44C66-C24 >0;
C33(Cll-C66)
C23 > 0 .
(1)
T h e s e conditions a r e well s a t i s f i e d by the moduli of table 1, the t h i r d r e l a t i o n being useful in r e j e c t i n g a s p u r i o u s root for C 13". I n t e r l a y e r bonds of the weak van a e r Waals type have been p r o p o s e d to account for the very easy cleavage of Bi2Te 3 and the soft acoustic modes, which c a u s e a d e p a r t u r e f r o m 7~ behavior in the low t e m p e r a t u r e specific heat [6]. The r e l a t i v e i m p o r t a n c e of c e n t r a l f o r c e s is indicated by the extent to which the Cauchy r e l a tions a r e satisfied. F o r t r i g o n a l s t r u c t u r e s these a r e given by: 3C66 = C l l ;
C44 = C 1 3 .
(2) 349
Volume 30A, number 6
PHYSICS
The present data deviate from obeying these e q u a l i t i e s by 5.7 and 7.5 p e r c e n t r e s p e c t i v e l y , s u g g e s t i n g t h a t the e l a s t i c b e h a v i o r is i n d e e d d o m i n a t e d by c e n t r a l van d e r W a a l s f o r c e s . A l a t t i c e d y n a m i c a l c a l c u l a t i o n is c u r r e n t l y b e i n g m a d e to e x p l o i t t h i s s i m p l e b o n d i n g s c h e m e . The limiting Debye temperature, obtained f r o m the m o d u l i at 4.2OK and u s i n g 167.17 ,~3 as the v o l u m e of the p r i m i t i v e c e l l , h a s b e e n c o m p u t e d by d i r e c t n u m e r i c a l i n t e g r a t i o n to be 158 + l ° K . T h i s r e s u l t a g r e e s with the c a l o r i m e t r i c v a l u e of 162 + 3OK, within the c o m b i n e d uncertainties.
LETTERS
17 November 1969
References 1. J . R . Drabble in P r o g r e s s in semiconductors, eds A. F. Gibson and R. E. Burgess, (Heywood and Co., London 1963) Vol. 7, p. 45. 2. D.I. Bolef and M. Menes, J. Appl. Phys. 31 {1960) 1010. 3. S. Epstein and A. P. de Bretteville J r . , Phys. Rev. 138 (1965) A 771. 4. Y. Ililavskii, Soy. Phys. Solid State 3 (1962) 2582. 5. R.B. Mallinson, J . A . Rayne and R. W. Ure J r . , Phys. Rev. 175 (1968) 1049. 6. G. Shoemake, J. Rayne and R. W. Ure J r . , Phys. Rev., to be published.
T h i s w o r k w a s s u p p o r t e d by a g r a n t f r o m the N a t i o n a l S c i e n c e Foundation.
P U L S E D S P I N - P R E C E S S I O N R E - O R I E N T A T I O N M E T H O D TO S T U D Y S P I N E X C H A N G E OF F R E E IONS * H. A. SCHUESSLER** Department of Physics, University of Washington, Seattle, Washington 98105, USA Received 16 October 1969
A scheme of consecutive square r.f. pulses was developed to study the electron spin re-orientation process of paramagnetic ions in spin exchange collisions with polarized atoms.
The high d e g r e e of i s o l a t i o n r e a l i z e d when s t o r i n g i o n s in a r. f. q u a d r u p o l e t r a p [1,2] s h o u l d m a k e a v a r i e t y of a t o m i c c o l l i s i o n e x p e r i m e n t s on c h a r g e d a t o m i c s y s t e m s p o s s i b l e . T h i s l e t t e r r e p o r t s the f i r s t m e a s u r e m e n t of an i o n - a t o m spin exchange time. Stored polarized ions were s u b j e c t e d to a s e q u e n c e of s h o r t r e s o n a n t r.f. p u l s e s , f o r e a c h of w h i c h the p u l s e l e n g t h Td and the p u l s e h e i g h t H 1 w e r e a d j u s t e d to p r o d u c e a 180 ° r o t a t i o n of the e l e c t r o n i c spin p o l a r i z a t i o n . T h e p u l s e p a r a m e t e r s then f u l f i l l the c o n d i t i o n s w 1 T d = z H 1 T d = II, T d < Te. A p o l a r i z e d a t o m i c b e a m p a s s i n g t h r o u g h the t r a p i n t e r a c t e d with the s t o r e d ions. T h e s p i n e x c h a n g e p r o c e s s c h a r a c t e r i z e d by the spin e x c h a n g e t i m e T e p a r t l y r e o r i e n t s the i n v e r t e d e l e c t r o n i c s p i n s of the i o n s d u r i n g the long r.f. f r e e p e r i o d s Tp b e t w e e n the * Supported by the U.S. Army Research Office (Durham). and by the National Science Foundation. ** Present Address: Department of Physics, Texas A and M University, College Station, Texas 77843. 350
s h o r t 180 ° p u l s e s . T e m a y be s t u d i e d by o b s e r v ing the t i m e e v o l u t i o n of the p o l a r i z a t i o n as a f u n c t i o n of T_. The n o v e l P s c h e m e w a s f i r s t u s e d to study the spin e x c h a n g e p r o c e s s CsT + 3He+~ ~ Cs~ + 3He+~
(1)
A l t h o u g h d u r i n g s p i n e x c h a n g e i t s e l f the n u c l e u s is not i m m e d i a t e l y a f f e c t e d , a n g u l a r m o m e n t u m is s u b s e q u e n t l y t r a n s f e r r e d to the n u c l e u s by the hfs i n t e r a c t i o n A I ' S and two e x c h a n g e c o l l i s i o n s a r e n e c e s s a r y to o r i e n t 3He+. The i n t e r e s t i n g Iris Z e e m a n l e v e l s a r e ( 1 , 1 ) ~ (1,0) at Vd, (I,i) ~-~ (i,-I) at re, and (i,0)~-~ (I,-i) at vf. °He+ ions were created inside a r.f. quadrupole trap by,pulsed electron bombardment of the background ~He gas, and the stored ions of an energy of a few eV were subjected to an optically pumped beam of thermal Cs atoms. In the resulting collisions, the ions quickly assumed the polarization of the beam by spin exchange. Their polarization was then inverted in each 180o pulse. The